cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-30 of 410 results. Next

A000041 a(n) is the number of partitions of n (the partition numbers).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of nonnegative solutions to b + 2c + 3d + 4e + ... = n and the number of nonnegative solutions to 2c + 3d + 4e + ... <= n. - Henry Bottomley, Apr 17 2001
a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).
Also the number of rooted trees with n+1 nodes and height at most 2.
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras gl(n). A006950, A015128 and this sequence together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). - Lekraj Beedassy, Oct 16 2004
Number of graphs on n vertices that do not contain P3 as an induced subgraph. - Washington Bomfim, May 10 2005
Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - Thomas Baruchel, Nov 07 2005
Sequence agrees with expansion of Molien series for symmetric group S_n up to the term in x^n. - Maurice D. Craig (towenaar(AT)optusnet.com.au), Oct 30 2006
Also the number of nonnegative integer solutions to x_1 + x_2 + x_3 + ... + x_n = n such that n >= x_1 >= x_2 >= x_3 >= ... >= x_n >= 0, because by letting y_k = x_k - x_(k+1) >= 0 (where 0 < k < n) we get y_1 + 2y_2 + 3y_3 + ... + (n-1)y_(n-1) + nx_n = n. - Werner Grundlingh (wgrundlingh(AT)gmail.com), Mar 14 2007
Let P(z) := Sum_{j>=0} b_j z^j, b_0 != 0. Then 1/P(z) = Sum_{j>=0} c_j z^j, where the c_j must be computed from the infinite triangular system b_0 c_0 = 1, b_0 c_1 + b_1 c_0 = 0 and so on (Cauchy products of the coefficients set to zero). The n-th partition number arises as the number of terms in the numerator of the expression for c_n: The coefficient c_n of the inverted power series is a fraction with b_0^(n+1) in the denominator and in its numerator having a(n) products of n coefficients b_i each. The partitions may be read off from the indices of the b_i. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007
A sequence of positive integers p = p_1 ... p_k is a descending partition of the positive integer n if p_1 + ... + p_k = n and p_1 >= ... >= p_k. If formally needed p_j = 0 is appended to p for j > k. Let P_n denote the set of these partition for some n >= 1. Then a(n) = 1 + Sum_{p in P_n} floor((p_1-1)/(p_2+1)). (Cf. A000065, where the formula reduces to the sum.) Proof in Kelleher and O'Sullivan (2009). For example a(6) = 1 + 0 + 0 + 0 + 0 + 1 + 0 + 0 + 1 + 1 + 2 + 5 = 11. - Peter Luschny, Oct 24 2010
Let n = Sum( k_(p_m) p_m ) = k_1 + 2k_2 + 5k_5 + 7k_7 + ..., where p_m is the m-th generalized pentagonal number (A001318). Then a(n) is the sum over all such pentagonal partitions of n of (-1)^(k_5+k_7 + k_22 + ...) ( k_1 + k_2 + k_5 + ...)! /( k_1! k_2! k_5! ...), where the exponent of (-1) is the sum of all the k's corresponding to even-indexed GPN's. - Jerome Malenfant, Feb 14 2011
From Jerome Malenfant, Feb 14 2011: (Start)
The matrix of a(n) values
a(0)
a(1) a(0)
a(2) a(1) a(0)
a(3) a(2) a(1) a(0)
....
a(n) a(n-1) a(n-2) ... a(0)
is the inverse of the matrix
1
-1 1
-1 -1 1
0 -1 -1 1
....
-d_n -d_(n-1) -d_(n-2) ... -d_1 1
where d_q = (-1)^(m+1) if q = m(3m-1)/2 = the m-th generalized pentagonal number (A001318), = 0 otherwise. (End)
Let k > 0 be an integer, and let i_1, i_2, ..., i_k be distinct integers such that 1 <= i_1 < i_2 < ... < i_k. Then, equivalently, a(n) equals the number of partitions of N = n + i_1 + i_2 + ... + i_k in which each i_j (1 <= j <= k) appears as a part at least once. To see this, note that the partitions of N of this class must be in 1-to-1 correspondence with the partitions of n, since N - i_1 - i_2 - ... - i_k = n. - L. Edson Jeffery, Apr 16 2011
a(n) is the number of distinct degree sequences over all free trees having n + 2 nodes. Take a partition of the integer n, add 1 to each part and append as many 1's as needed so that the total is 2n + 2. Now we have a degree sequence of a tree with n + 2 nodes. Example: The partition 3 + 2 + 1 = 6 corresponds to the degree sequence {4, 3, 2, 1, 1, 1, 1, 1} of a tree with 8 vertices. - Geoffrey Critzer, Apr 16 2011
a(n) is number of distinct characteristic polynomials among n! of permutations matrices size n X n. - Artur Jasinski, Oct 24 2011
Conjecture: starting with offset 1 represents the numbers of ordered compositions of n using the signed (++--++...) terms of A001318 starting (1, 2, -5, -7, 12, 15, ...). - Gary W. Adamson, Apr 04 2013 (this is true by the pentagonal number theorem, Joerg Arndt, Apr 08 2013)
a(n) is also number of terms in expansion of the n-th derivative of log(f(x)). In Mathematica notation: Table[Length[Together[f[x]^n * D[Log[f[x]], {x, n}]]], {n, 1, 20}]. - Vaclav Kotesovec, Jun 21 2013
Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - Zhi-Wei Sun, Dec 02 2013
Partitions of n that contain a part p are the partitions of n - p. Thus, number of partitions of m*n - r that include k*n as a part is A000041(h*n-r), where h = m - k >= 0, n >= 2, 0 <= r < n; see A111295 as an example. - Clark Kimberling, Mar 03 2014
a(n) is the number of compositions of n into positive parts avoiding the pattern [1, 2]. - Bob Selcoe, Jul 08 2014
Conjecture: For any j there exists k such that all primes p <= A000040(j) are factors of one or more a(n) <= a(k). Growth of this coverage is slow and irregular. k = 1067 covers the first 102 primes, thus slower than A000027. - Richard R. Forberg, Dec 08 2014
a(n) is the number of nilpotent conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015
Define a segmented partition a(n,k, ) to be a partition of n with exactly k parts, with s(j) parts t(j) identical to each other and distinct from all the other parts. Note that n >= k, j <= k, 0 <= s(j) <= k, s(1)t(1) + ... + s(j)t(j) = n and s(1) + ... + s(j) = k. Then there are up to a(k) segmented partitions of n with exactly k parts. - Gregory L. Simay, Nov 08 2015
(End)
From Gregory L. Simay, Nov 09 2015: (Start)
The polynomials for a(n, k, ) have degree j-1.
a(n, k, ) = 1 if n = 0 mod k, = 0 otherwise
a(rn, rk, ) = a(n, k, )
a(n odd, k, ) = 0
Established results can be recast in terms of segmented partitions:
For j(j+1)/2 <= n < (j+1)(j+2)/2, A000009(n) = a(n, 1, <1>) + ... + a(n, j, ), j < n
a(n, k, ) = a(n - j(j-1)/2, k)
(End)
a(10^20) was computed using the NIST Arb package. It has 11140086260 digits and its head and tail sections are 18381765...88091448. See the Johansson 2015 link. - Stanislav Sykora, Feb 01 2016
Satisfies Benford's law [Anderson-Rolen-Stoehr, 2011]. - N. J. A. Sloane, Feb 08 2017
The partition function p(n) is log-concave for all n>25 [DeSalvo-Pak, 2014]. - Michel Marcus, Apr 30 2019
a(n) is also the dimension of the n-th cohomology of the infinite real Grassmannian with coefficients in Z/2. - Luuk Stehouwer, Jun 06 2021
Number of equivalence relations on n unlabeled nodes. - Lorenzo Sauras Altuzarra, Jun 13 2022
Equivalently, number of idempotent mappings f from a set X of n elements into itself (i.e., satisfying f o f = f) up to permutation (i.e., f~f' :<=> There is a permutation sigma in Sym(X) such that f' o sigma = sigma o f). - Philip Turecek, Apr 17 2023
Conjecture: Each integer n > 2 different from 6 can be written as a sum of finitely many numbers of the form a(k) + 2 (k > 0) with no summand dividing another. This has been verified for n <= 7140. - Zhi-Wei Sun, May 16 2023
a(n) is also the number of partitions of n*(n+3)/2 into n distinct parts. - David García Herrero, Aug 20 2024
a(n) is also the number of non-isomorphic sigma algebras on {1,...,n}. A000110(n) counts all sigma algebras on {1,...,n}. Every sigma algebra on a finite set X is exactly the collection of all unions of its atoms (its minimal nonempty members), and those atoms partition X. An isomorphism of sigma algebras must map atoms to atoms, so the isomorphism class of a sigma algebra is determined by the multiset of its atom-sizes, which is an integer partition of n. - Matthew Azar, Jul 18 2025

Examples

			a(5) = 7 because there are seven partitions of 5, namely: {1, 1, 1, 1, 1}, {2, 1, 1, 1}, {2, 2, 1}, {3, 1, 1}, {3, 2}, {4, 1}, {5}. - _Bob Selcoe_, Jul 08 2014
G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = 1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ...
From _Gregory L. Simay_, Nov 08 2015: (Start)
There are up to a(4)=5 segmented partitions of the partitions of n with exactly 4 parts. They are a(n,4, <4>), a(n,4,<3,1>), a(n,4,<2,2>), a(n,4,<2,1,1>), a(n,4,<1,1,1,1>).
The partition 8,8,8,8 is counted in a(32,4,<4>).
The partition 9,9,9,5 is counted in a(32,4,<3,1>).
The partition 11,11,5,5 is counted in a(32,4,<2,2>).
The partition 13,13,5,1 is counted in a(32,4,<2,1,1>).
The partition 14,9,6,3 is counted in a(32,4,<1,1,1,1>).
a(n odd,4,<2,2>) = 0.
a(12, 6, <2,2,2>) = a(6,3,<1,1,1>) = a(6-3,3) = a(3,3) = 1. The lone partition is 3,3,2,2,1,1.
(End)

References

  • George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
  • George E. Andrews and K. Ericksson, Integer Partitions, Cambridge University Press 2004.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 307.
  • R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III.
  • Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
  • Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
  • Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag.
  • B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 999.
  • J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 183.
  • Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 411.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 94-96.
  • L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp. 101-164, Chelsea NY 1992.
  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 37, Eq. (22.13).
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
  • G. H. Hardy and S. Ramanujan, Asymptotic formulas in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-.
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 83-100, 113-131.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, 273-296.
  • D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 396.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.1, p. 491.
  • S. Ramanujan, Collected Papers, Chap. 25, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1919), pp. 207-213).
  • S. Ramanujan, Collected Papers, Chap. 28, Cambridge Univ. Press 1927 (Proceedings of the London Math. Soc., 2, 18(1920)).
  • S. Ramanujan, Collected Papers, Chap. 30, Cambridge Univ. Press 1927 (Mathematische Zeitschrift, 9 (1921), pp. 147-163).
  • S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Table IV on page 308.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 122.
  • J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. E. Tapscott and D. Marcovich, "Enumeration of Permutational Isomers: The Porphyrins", Journal of Chemical Education, 55 (1978), 446-447.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 286-289, 297-298, 303.
  • Robert M. Young, "Excursions in Calculus", Mathematical Association of America, p. 367.

Links

Crossrefs

Cf. A000009, A000079, A000203, A001318, A008284, A026820, A065446, A078506, A113685, A132311, A000248, A000110.
Partial sums give A000070.
For successive differences see A002865, A053445, A072380, A081094, A081095.
Antidiagonal sums of triangle A092905. a(n) = A054225(n,0).
Boustrophedon transforms: A000733, A000751.
Cf. A167376 (complement), A061260 (multisets), A000700 (self-conjug), A330644 (not self-conj).

Programs

  • GAP
    List([1..10],n->Size(OrbitsDomain(SymmetricGroup(IsPermGroup,n),SymmetricGroup(IsPermGroup,n),\^))); # Attila Egri-Nagy, Aug 15 2014
  • Haskell
    import Data.MemoCombinators (memo2, integral)
a000041 n = a000041_list !! n
a000041_list = map (p' 1) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < k then 0 else p' k (m - k) + p' (k + 1) m
-- Reinhard Zumkeller, Nov 03 2015, Nov 04 2013
  • Julia
    # DedekindEta is defined in A000594
A000041List(len) = DedekindEta(len, -1)
A000041List(50) |> println # Peter Luschny, Mar 09 2018
  • Magma
    a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];
  • Maple
    A000041 := n -> combinat:-numbpart(n): [seq(A000041(n), n=0..50)]; # Warning: Maple 10 and 11 give incorrect answers in some cases: A110375.
spec := [B, {B=Set(Set(Z,card>=1))}, unlabeled ];
[seq(combstruct[count](spec, size=n), n=0..50)];
with(combstruct):ZL0:=[S,{S=Set(Cycle(Z,card>0))}, unlabeled]: seq(count(ZL0,size=n),n=0..45); # Zerinvary Lajos, Sep 24 2007
G:={P=Set(Set(Atom,card>0))}: combstruct[gfsolve](G,labeled,x); seq(combstruct[count]([P,G,unlabeled],size=i),i=0..45); # Zerinvary Lajos, Dec 16 2007
# Using the function EULER from Transforms (see link at the bottom of the page).
1,op(EULER([seq(1,n=1..49)])); # Peter Luschny, Aug 19 2020
  • Mathematica
    Table[ PartitionsP[n], {n, 0, 45}]
a[ n_] := SeriesCoefficient[ q^(1/24) / DedekindEta[ Log[q] / (2 Pi I)], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)
CoefficientList[1/QPochhammer[q] + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015 *)
a[0] := 1; a[n_] := a[n] = Block[{k=1, s=0, i=n-1}, While[i >= 0, s=s-(-1)^k (a[i]+a[i-k]); k=k+1; i=i-(3 k-2)]; s]; Map[a, Range[0, 49]] (* Oliver Seipel, Jun 01 2024 after Euler *)
  • Maxima
    num_partitions(60,list); /* Emanuele Munarini, Feb 24 2014 */
  • MuPAD
    combinat::partitions::count(i) $i=0..54 // Zerinvary Lajos, Apr 16 2007
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x * O(x^n)), n))};
  • PARI
    /* The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): */
Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))
L(n, q) = if(q==1,1,sum(h=1,q-1,if(gcd(h,q)>1,0,cos((g(h,q)-2*h*n)*Pi/q))))
g(h, q) = if(q<3,0,sum(k=1,q-1,k*(frac(h*k/q)-1/2)))
part(n) = round(sum(q=1,max(5,0.5*sqrt(n)),L(n,q)*Psi(n,q)))
/* Ralf Stephan, Nov 30 2002, fixed by Vaclav Kotesovec, Apr 09 2018 */
  • PARI
    {a(n) = numbpart(n)};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), x^k^2 / prod( i=1, k, 1 - x^i, 1 + x * O(x^n))^2, 1), n))};
  • PARI
    f(n)= my(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]1,i--;s+=i*(v[i]=(n-s)\i));t++);t \\ Thomas Baruchel, Nov 07 2005
  • PARI
    a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010
  • Perl
    use ntheory ":all"; my @p = map { partitions($) } 0..100; say "[@p]"; # _Dana Jacobsen, Sep 06 2015
  • Python
    from sympy.functions.combinatorial.numbers import partition
print([partition(i) for i in range(101)]) # Joan Ludevid, May 25 2025
  • Racket
    #lang racket
; SUM(k,-inf,+inf) (-1)^k p(n-k(3k-1)/2)
; For k outside the range (1-(sqrt(1-24n))/6 to (1+sqrt(1-24n))/6) argument n-k(3k-1)/2 < 0.
; Therefore the loops below are finite. The hash avoids repeated identical computations.
(define (p n) ; Nr of partitions of n.
(hash-ref h n
  (λ ()
   (define r
    (+
     (let loop ((k 1) (n (sub1 n)) (s 0))
      (if (< n 0) s
       (loop (add1 k) (- n (* 3 k) 1) (if (odd? k) (+ s (p n)) (- s (p n))))))
     (let loop ((k -1) (n (- n 2)) (s 0))
      (if (< n 0) s
       (loop (sub1 k) (+ n (* 3 k) -2) (if (odd? k) (+ s (p n)) (- s (p n))))))))
   (hash-set! h n r)
   r)))
(define h (make-hash '((0 . 1))))
; (for ((k (in-range 0 50))) (printf "~s, " (p k))) runs in a moment.
; Jos Koot, Jun 01 2016
  • Sage
    [number_of_partitions(n) for n in range(46)]  # Zerinvary Lajos, May 24 2009
  • Sage
    @CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[A000041(n) for n in range(50)]  # Peter Luschny, Oct 13 2012
  • Sage
    # uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(1, 0)
b = EulerTransform(a)
print([b(n) for n in range(50)]) # Peter Luschny, Nov 11 2020

Formula

G.f.: Product_{k>0} 1/(1-x^k) = Sum_{k>= 0} x^k Product_{i = 1..k} 1/(1-x^i) = 1 + Sum_{k>0} x^(k^2)/(Product_{i = 1..k} (1-x^i))^2.
G.f.: 1 + Sum_{n>=1} x^n/(Product_{k>=n} 1-x^k). - Joerg Arndt, Jan 29 2011
a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + ... = 0, where the sum is over n-k and k is a generalized pentagonal number (A001318) <= n and the sign of the k-th term is (-1)^([(k+1)/2]). See A001318 for a good way to remember this!
a(n) = (1/n) * Sum_{k=0..n-1} sigma(n-k)*a(k), where sigma(k) is the sum of divisors of k (A000203).
a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy and Ramanujan). See A050811.
a(n) = a(0)*b(n) + a(1)*b(n-2) + a(2)*b(n-4) + ... where b = A000009.
From Jon E. Schoenfield, Aug 17 2014: (Start)
It appears that the above approximation from Hardy and Ramanujan can be refined as
a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3 + c0 + c1/n^(1/2) + c2/n + c3/n^(3/2) + c4/n^2 + ...)), where the coefficients c0 through c4 are approximately
c0 = -0.230420145062453320665537
c1 = -0.0178416569128570889793
c2 = 0.0051329911273
c3 = -0.0011129404
c4 = 0.0009573,
as n -> infinity. (End)
From Vaclav Kotesovec, May 29 2016 (c4 added Nov 07 2016): (Start)
c0 = -0.230420145062453320665536704197233... = -1/36 - 2/Pi^2
c1 = -0.017841656912857088979502135349949... = 1/(6*sqrt(6)*Pi) - sqrt(3/2)/Pi^3
c2 = 0.005132991127342167594576391633559... = 1/(2*Pi^4)
c3 = -0.001112940489559760908236602843497... = 3*sqrt(3/2)/(4*Pi^5) - 5/(16*sqrt(6)*Pi^3)
c4 = 0.000957343284806972958968694349196... = 1/(576*Pi^2) - 1/(24*Pi^4) + 93/(80*Pi^6)
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n) + (1/16 + Pi^2/6912)/n).
a(n) ~ exp(Pi*sqrt(2*n/3) - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n) + (1/24 - 3/(4*Pi^2))/n) / (4*sqrt(3)*n).
(End)
a(n) < exp( (2/3)^(1/2) Pi sqrt(n) ) (Ayoub, p. 197).
G.f.: Product_{m>=1} (1+x^m)^A001511(m). - Vladeta Jovovic, Mar 26 2004
a(n) = Sum_{i=0..n-1} P(i, n-i), where P(x, y) is the number of partitions of x into at most y parts and P(0, y)=1. - Jon Perry, Jun 16 2003
G.f.: Product_{i>=1} Product_{j>=0} (1+x^((2i-1)*2^j))^(j+1). - Jon Perry, Jun 06 2004
G.f. e^(Sum_{k>0} (x^k/(1-x^k)/k)). - Franklin T. Adams-Watters, Feb 08 2006
a(n) = A114099(9*n). - Reinhard Zumkeller, Feb 15 2006
Euler transform of all 1's sequence (A000012). Weighout transform of A001511. - Franklin T. Adams-Watters, Mar 15 2006
a(n) = A027187(n) + A027193(n) = A000701(n) + A046682(n). - Reinhard Zumkeller, Apr 22 2006
A026820(a(n),n) = A134737(n) for n > 0. - Reinhard Zumkeller, Nov 07 2007
Convolved with A152537 gives A000079, powers of 2. - Gary W. Adamson, Dec 06 2008
a(n) = A026820(n, n); a(n) = A108949(n) + A045931(n) + A108950(n) = A130780(n) + A171966(n) - A045931(n) = A045931(n) + A171967(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = Tr(n)/(24*n-1) = A183011(n)/A183010(n), n>=1. See the Bruinier-Ono paper in the Links. - Omar E. Pol, Jan 23 2011
From Jerome Malenfant, Feb 14 2011: (Start)
a(n) = determinant of the n X n Toeplitz matrix:
1 -1
1 1 -1
0 1 1 -1
0 0 1 1 -1
-1 0 0 1 1 -1
. . .
d_n d_(n-1) d_(n-2)...1
where d_q = (-1)^(m+1) if q = m(3m-1)/2 = p_m, the m-th generalized pentagonal number (A001318), otherwise d_q = 0. Note that the 1's run along the diagonal and the -1's are on the superdiagonal. The (n-1) row (not written) would end with ... 1 -1. (End)
Empirical: let F*(x) = Sum_{n=0..infinity} p(n)*exp(-Pi*x*(n+1)), then F*(2/5) = 1/sqrt(5) to a precision of 13 digits.
F*(4/5) = 1/2+3/2/sqrt(5)-sqrt(1/2*(1+3/sqrt(5))) to a precision of 28 digits. These are the only values found for a/b when a/b is from F60, Farey fractions up to 60. The number for F*(4/5) is one of the real roots of 25*x^4 - 50*x^3 - 10*x^2 - 10*x + 1. Note here the exponent (n+1) compared to the standard notation with n starting at 0. - Simon Plouffe, Feb 23 2011
The constant (2^(7/8)*GAMMA(3/4))/(exp(Pi/6)*Pi^(1/4)) = 1.0000034873... when expanded in base exp(4*Pi) will give the first 52 terms of a(n), n>0, the precision needed is 300 decimal digits. - Simon Plouffe, Mar 02 2011
a(n) = A035363(2n). - Omar E. Pol, Nov 20 2009
G.f.: A(x)=1+x/(G(0)-x); G(k) = 1 + x - x^(k+1) - x*(1-x^(k+1))/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 25 2012
Convolution of A010815 with A000712. - Gary W. Adamson, Jul 20 2012
G.f.: 1 + x*(1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2013
G.f.: Q(0) where Q(k) = 1 + x^(4*k+1)/( (x^(2*k+1)-1)^2 - x^(4*k+3)*(x^(2*k+1)-1)^2/( x^(4*k+3) + (x^(2*k+2)-1)^2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 16 2013
a(n) = 24*spt(n) + 12*N_2(n) - Tr(n) = 24*A092269(n) + 12*A220908(n) - A183011(n), n >= 1. - Omar E. Pol, Feb 17 2013
G.f.: 1/(x; x){inf} where (a; q)_k is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Apr 24 2013
a(n) = A066186(n)/n, n >= 1. - Omar E. Pol, Aug 16 2013
From Peter Bala, Dec 23 2013: (Start)
a(n-1) = Sum_{parts k in all partitions of n} mu(k), where mu(k) is the arithmetical Möbius function (see A008683).
Let P(2,n) denote the set of partitions of n into parts k >= 2. Then a(n-2) = -Sum_{parts k in all partitions in P(2,n)} mu(k).
n*( a(n) - a(n-1) ) = Sum_{parts k in all partitions in P(2,n)} k (see A138880).
Let P(3,n) denote the set of partitions of n into parts k >= 3. Then
a(n-3) = (1/2)*Sum_{parts k in all partitions in P(3,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, we can find an approximate 3-term recurrence for the partition function: a(n) ~ a(n-1) + a(n-2) + (Pi^2/(3*n) - 1)*a(n-3). For example, substituting the values a(47) = 124754, a(48) = 147273 and a(49) = 173525 into the recurrence gives the approximation a(50) ~ 204252.48... compared with the true value a(50) = 204226. (End)
a(n) = Sum_{k=1..n+1} (-1)^(n+1-k)*A000203(k)*A002040(n+1-k). - Mircea Merca, Feb 27 2014
a(n) = A240690(n) + A240690(n+1), n >= 1. - Omar E. Pol, Mar 16 2015
From Gary W. Adamson, Jun 22 2015: (Start)
A production matrix for the sequence with offset 1 is M, an infinite n x n matrix of the following form:
a, 1, 0, 0, 0, 0, ...
b, 0, 1, 0, 0, 0, ...
c, 0, 0, 1, 0, 0, ...
d, 0, 0, 0, 1, 0, ...
.
.
... such that (a, b, c, d, ...) is the signed version of A080995 with offset 1: (1,1,0,0,-1,0,-1,...)
and a(n) is the upper left term of M^n.
This operation is equivalent to the g.f. (1 + x + 2x^2 + 3x^3 + 5x^4 + ...) = 1/(1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + ...). (End)
G.f.: x^(1/24)/eta(log(x)/(2 Pi i)). - Thomas Baruchel, Jan 09 2016, after Michael Somos (after Richard Dedekind).
a(n) = Sum_{k=-inf..+inf} (-1)^k a(n-k(3k-1)/2) with a(0)=1 and a(negative)=0. The sum can be restricted to the (finite) range from k = (1-sqrt(1-24n))/6 to (1+sqrt(1-24n))/6, since all terms outside this range are zero. - Jos Koot, Jun 01 2016
G.f.: (conjecture) (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) is A000009: (1, 1, 1, 2, 2, 3, 4, ...). - Gary W. Adamson, Sep 18 2016; Doron Zeilberger observed today that "This follows immediately from Euler's formula 1/(1-z) = (1+z)*(1+z^2)*(1+z^4)*(1+z^8)*..." Gary W. Adamson, Sep 20 2016
a(n) ~ 2*Pi * BesselI(3/2, sqrt(24*n-1)*Pi/6) / (24*n-1)^(3/4). - Vaclav Kotesovec, Jan 11 2017
G.f.: Product_{k>=1} (1 + x^k)/(1 - x^(2*k)). - Ilya Gutkovskiy, Jan 23 2018
a(n) = p(1, n) where p(k, n) = p(k+1, n) + p(k, n-k) if k < n, 1 if k = n, and 0 if k > n. p(k, n) is the number of partitions of n into parts >= k. - Lorraine Lee, Jan 28 2020
Sum_{n>=1} 1/a(n) = A078506. - Amiram Eldar, Nov 01 2020
Sum_{n>=0} a(n)/2^n = A065446. - Amiram Eldar, Jan 19 2021
From Simon Plouffe, Mar 12 2021: (Start)
Sum_{n>=0} a(n)/exp(Pi*n) = 2^(3/8)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/24)).
Sum_{n>=0} a(n)/exp(2*Pi*n) = 2^(1/2)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/12)).
[corrected by Vaclav Kotesovec, May 12 2023] (End)
[These are the reciprocals of phi(exp(-Pi)) (A259148) and phi(exp(-2*Pi)) (A259149), where phi(q) is the Euler modular function. See B. C. Berndt (RLN, Vol. V, p. 326), and formulas (13) and (14) in I. Mező, 2013. - Peter Luschny, Mar 13 2021]
a(n) = A000009(n) + A035363(n) + A006477(n). - R. J. Mathar, Feb 01 2022
a(n) = A008284(2*n,n) is also the number of partitions of 2n into n parts. - Ryan Brooks, Jun 11 2022
a(n) = A000700(n) + A330644(n). - R. J. Mathar, Jun 15 2022
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 + Sum_{r>=1} w(r)/n^(r/2)), where w(r) = 1/(-4*sqrt(6))^r * Sum_{k=0..(r+1)/2} binomial(r+1,k) * (r+1-k) / (r+1-2*k)! * (Pi/6)^(r-2*k) [Cormac O'Sullivan, 2023, pp. 2-3]. - Vaclav Kotesovec, Mar 15 2023

Extensions

Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001
Additional comments from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

When convolved with the partition numbers A000041 gives 1, 0, 0, 0, 0, ...
Also, number of different partitions of n into parts of -1 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
The comment that "when convolved with the partition numbers gives [1, 0, 0, 0, ...]" is equivalent to row sums of triangle A145975 = [1, 0, 0, 0, ...]; where A145975 is a partition number convolution triangle. - Gary W. Adamson, Oct 25 2008
When convolved with n-th partial sums of A000041 = the binomial sequence starting (1, n, ...). Example: A010815 convolved with A014160 (partial sum operation applied thrice to the partition numbers) = (1, 3, 6, 10, ...). - Gary W. Adamson, Nov 11 2008
(A000012^(-n) * A000041) convolved with A010815 = n-th row of the inverse of Pascal's triangle, (as a vector, followed by zeros); where A000012^(-1) = the pairwise difference operator. Example: (A000012^(-4) * A000041) convolved with A010815 = (1, -4, 6, -4, 1, 0, 0, 0, ...). - Gary W. Adamson, Nov 11 2008
Also sum of [product of (1-2/(hook lengths)^2)] over all partitions of n. - Wouter Meeussen, Sep 16 2010
Cayley (1895) begins article 387 with "Write for shortness sqrt(2k'K / pi) / [1-q^{2m-1}]^2 = G, ..." which is a convoluted way of writing G = [1-q^{2m}] = (1-q^2)(1-q^4)... - Michael Somos, Aug 01 2011
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^3, b = x. - Michael Somos, Jan 21 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 1 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 - x^40 + ...
G.f. = q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 + q^625 + ...
From _Seiichi Manyama_, Mar 04 2017: (Start)
G.f.
= 1 + (-x - 3*x^2/2 - 4*x^3/3 -  7*x^4/4  -  6*x^5/5 - ...)
     + 1/2 * (x^2   + 3*x^3   + 59*x^4/12 + 15*x^5/2 + ...)
              + 1/6 * (-x^3   -  9*x^4/2  - 43*x^5/4 - ...)
                         + 1/24 * (x^4    +  6*x^5   + ...)
                                   + 1/120 * (-x^5   - ...)
                                             + ...
= 1 - x - x^2 + x^5 + .... (End)

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 825.
  • B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054. See page 3.
  • T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.
  • A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 295, Art. 387.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5g].
  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.12) and (32.13).
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 86.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 353.
  • B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 70.
  • A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 186.

Links

Crossrefs

Cf. A000041, A001318 (characteristic function), A000326, A080995.
Cf. A067659, A067661.
Cf. A145975, A002865, A014160.
Cf. also A170925, A143374, A194087, A242168, A258232, A143062, A203568, A002111, A380038.

Programs

  • Julia
    # DedekindEta is defined in A000594.
A010815List(len) = DedekindEta(len, 1)
A010815List(93) |> println # Peter Luschny, Mar 09 2018
  • Julia
    function A010815(n)
    r = 24 * n + 1
    m = isqrt(r)
    m * m != r && return 0
    iseven(div(m + m % 6, 6)) ? 1 : -1
end # Peter Luschny, Sep 09 2021
  • Magma
    Coefficients(&*[1-x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Vincenzo Librandi, Jan 15 2017
  • Maple
    A010815 := mul((1-x^m), m=1..100);
A010815 := proc(n) local x,m;
    product(1-x^m,m=1..n) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016
A010815 := proc(n) 24*n + 1; if issqr(%) then sqrt(%);
(-1)^irem(iquo(% + irem(%, 6), 6), 2) else 0 fi end: # Peter Luschny, Oct 02 2022
  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> -x^(1/2)}, {x, 0, n}]]; (* Michael Somos, Nov 15 2011 *)
CoefficientList[ Series[ Product[(1 - x^k), {k, 1, 70}], {x, 0, 70}], x]
(* hooklength[ ] cfr A047874 *) Table[ Tr[ ( Times@@(1-2/Flatten[hooklength[ # ]]^2) )&/@ Partitions[n] ],{n,26}] (* Wouter Meeussen, Sep 16 2010 *)
CoefficientList[ Series[ QPochhammer[q], {q, 0, 100}], q] (* Jean-François Alcover, Dec 04 2013 *)
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ[m], KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Jun 04 2015 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = -1; Do[Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, May 04 2018 *)
Table[m = (1 + Sqrt[1 + 24*k])/6; If[IntegerQ[m], (-1)^m, 0] + If[IntegerQ[m - 1/3], (-1)^(m - 1/3), 0], {k, 0, 100}] (* Vaclav Kotesovec, Jul 09 2020 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n)), n))}; /* Michael Somos, Jun 05 2002 */
  • PARI
    {a(n) = polcoeff( prod( k=1, n, 1 - x^k, 1 + x * O(x^n)), n)}; /* Michael Somos, Nov 19 2011 */
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n))}; /* Michael Somos, Feb 26 2006 */
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), if( (n%2) && (n%3), (-1)^round( n/6 )))}; /* Michael Somos, Feb 26 2006 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = 1 + O(x^n); polcoeff( sum( k=1, (sqrtint( 8*n + 1)-1) \ 2, A *= x^k / (x^k - 1) + x * O(x^(n - (k^2-k)/2)), 1), n))}; /* Michael Somos, Aug 18 2006 */
  • PARI
    lista(nn) = {q='q+O('q^nn); Vec(eta(q))} \\ Altug Alkan, Mar 21 2018
  • Python
    from math import isqrt
def A010815(n):
    m = isqrt(24*n+1)
    return 0 if m**2 != 24*n+1 else ((-1)**((m-1)//6) if m % 6 == 1 else (-1)**((m+1)//6)) # Chai Wah Wu, Sep 08 2021

Formula

a(n) = (-1)^m if n is of the form m(3m+-1)/2; otherwise a(n)=0. The values of n such that |a(n)|=1 are the generalized pentagonal numbers, A001318. The values of n such that a(n)=0 is A090864.
Expansion of the Dedekind eta function without the q^(1/24) factor in powers of q.
Euler transform of period 1 sequence [ -1, -1, -1, ...].
G.f.: (q; q){oo} = Product{k >= 1} (1-q^k) = Sum_{n=-oo..oo} (-1)^n*q^(n*(3n+1)/2). The first notation is a q-Pochhammer symbol.
Expansion of f(-x) := f(-x, -x^2) in powers of x. A special case of Ramanujan's general theta function; see Berndt reference. - Michael Somos, Apr 08 2003
a(n) = A067661(n) - A067659(n). - Jon Perry, Jun 17 2003
Expansion of f(x^5, x^7) - x * f(x, x^11) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 21 2012
G.f.: q^(-1/24) * eta(t), where q = exp(2 Pi i t) and eta is the Dedekind eta function.
G.f.: 1 - x - x^2(1-x) - x^3(1-x)(1-x^2) - ... - Jon Perry, Aug 07 2004
Given g.f. A(x), then B(q) = q * A(q^3)^8 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - v^3 + 16*u*w^2. - Michael Somos, May 02 2005
Given g.f. A(x), then B(q) = q * A(q^24) satisfies 0 = f(B(q), B(x^q), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^9*u3*u6^3 - u2^9*u3^4 + 9*u1^4*u2*u6^8. - Michael Somos, May 02 2005
a(n) = b(24*n + 1) where b() is multiplicative with b(p^2e) = (-1)^e if p == 5 or 7 (mod 12), b(p^2e) = +1 if p == 1 or 11 (mod 12) and b(p^(2e-1)) = b(2^e) = b(3^e) = 0 if e>0. - Michael Somos, May 08 2005
Given g.f. A(x), then B(q) = q * A(q^24) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^16*w^8 - v^24 + 16*u^8*w^16. - Michael Somos, May 08 2005
a(n) = (-1)^n * A121373(n). a(25*n + 1) = -a(n). a(5*n + 3) = a(5*n + 4) = 0. a(5*n) = A113681(n). a(5*n + 2) = - A116915(n). - Michael Somos, Feb 26 2006
G.f.: 1 + Sum_{k>0} (-1)^k * x^((k^2 + k) / 2) / ((1 - x) * (1 - x^2) * ... * (1 - x^k)). - Michael Somos, Aug 18 2006
a(n) = -(1/n)*Sum_{k=1..n} sigma(k)*a(n-k). - Vladeta Jovovic, Aug 28 2002
G.f.: A(x) = 1 - x/G(0); G(k) = 1 + x - x^(k+1) - x*(1-x^(k+1))/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2012
Expansion of f(-x^2) * chi(-x) = psi(-x) * chi(-x^2) = psi(x) * chi(-x)^2 = f(-x^2)^2 / psi(x) = phi(-x) / chi(-x) = phi(-x^2) / chi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Nov 16 2015
G.f.: exp( Sum_{n>=1} -sigma(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017
G.f.: Sum_{n >= 0} x^(n*(2*n-1))*(2*x^(2*n) - 1)/Product_{k = 1..2*n} 1 - x^k. - Peter Bala, Feb 02 2021
The g.f. A(x) satisfies A(x^2) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k >= n+1} 1 - x^k = 1 - x^2 - x^4 + x^10 + x^14 - x^24 - x^30 + + - - .... - Peter Bala, Feb 12 2021
For m >= 0, A(x) = (1 - x)*(1 - x^2)*...*(1 - x^m) * Sum_{n >= 0} (-1)^n * x^(n*(n+2*m+1)/2) /(Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025
From Friedjof Tellkamp, Mar 19 2025: (Start)
Sum_{n>=1} a(n)/n = 6 - 4*Pi/sqrt(3).
Sum_{n>=1} a(n)/n^2 = -108 + 16*sqrt(3)*Pi + 2*Pi^2.
Sum_{n>=1} a(n)/n^k = Sum_{i=0..k} 6^(k-i)*C(-k, k-i)*A(i), where A(i)=(2^i-2)*(3^i-3)*zeta(i) for even i, and A(i)=-G(i/2-1/2)*(2^i+2)*(2*Pi)^i/(sqrt(3)*Gamma(i+1)) for odd i, with G(n>0) as the Glaisher's numbers (A002111) and G(0)=1/2. (End)

Extensions

Additional comments from Michael Somos, Jun 05 2002

A001055 The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 9, 1, 5, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From David W. Wilson, Feb 28 2009: (Start)
By a factorization of n we mean a multiset of integers > 1 whose product is n.
For example, 6 is the product of 2 such multisets, {2, 3} and {6}, so a(6) = 2.
Similarly 8 is the product of 3 such multisets, {2, 2, 2}, {2, 4} and {8}, so a(8) = 3.
1 is the product of 1 such multiset, namely the empty multiset {}, whose product is by definition the multiplicative identity 1. Hence a(1) = 1. (End)
a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller, Sep 21 2001; Benoit Cloitre and N. J. A. Sloane, May 15 2002
Number of members of A025487 with n divisors. - Matthew Vandermast, Jul 12 2004
See sequence A162247 for a list of the factorizations of n and a program for generating the factorizations for any n. - T. D. Noe, Jun 28 2009
So a(n) gives the number of different prime signatures that can be found among the integers that have n divisors. - Michel Marcus, Nov 11 2015
For n > 0, also the number of integer partitions of n with product n, ranked by A301987. For example, the a(12) = 4 partitions are: (12), (6,2,1,1,1,1), (4,3,1,1,1,1,1), (3,2,2,1,1,1,1,1). See also A380218. In general, A379666(m,n) = a(n) for any m >= n. - Gus Wiseman, Feb 07 2025

Examples

			1: 1, a(1) = 1
2: 2, a(2) = 1
3: 3, a(3) = 1
4: 4 = 2*2, a(4) = 2
6: 6 = 2*3, a(6) = 2
8: 8 = 2*4 = 2*2*2, a(8) = 3
etc.

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
  • S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
  • Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 198, exercise 9 (in the third edition 2015, p. 296, exercise 211).

Links

Crossrefs

A045782 gives the range of a(n).
For records see A033833, A033834.
Cf. A002033, A045778 (strict version), A050322, A050336, A064553, A064554, A064555, A077565, A051731, A005171, A097296, A190938, A216599, A216600, A216601, A216602.
Row sums of A316439 (for n>1).
Cf. A096276 (partial sums).
The additive version is A000041 (integer partitions), strict A000009.
Row sums of A318950.
A002865 counts partitions into parts > 1.
A069016 counts distinct sums of factorizations.
A319000 counts partitions by product and sum, row sums A319916.
A379666 (transpose A380959) counts partitions by sum and product, without 1's A379668, strict A379671.
Cf. A003963, A057567, A057568 (strict A379733), A114324, A301987, A319005, A379734.

Programs

  • Haskell
    a001055 = (map last a066032_tabl !!) . (subtract 1)
-- Reinhard Zumkeller, Oct 01 2012
  • Java
    public class MultiPart {
    public static void main(String[] argV) {
        for (int i=1;i<=100;++i) System.out.println(1+getDivisors(2,i));
    }
    public static int getDivisors(int min,int n) {
        int total = 0;
        for (int i=min;i=i) { ++total; if (n/i>i) total+=getDivisors(i,n/i); }
        return total;
    }
} \\ Scott R. Shannon, Aug 21 2019
  • Maple
    with(numtheory):
T := proc(n::integer, m::integer)
        local A, summe, d:
        if isprime(n) then
                if n <= m then
                        return 1;
                end if:
                return 0 ;
        end if:
        A := divisors(n) minus {n, 1}:
        for d in A do
                if d > m then
                        A := A minus {d}:
                end if:
        end do:
        summe := add(T(n/d,d),d=A) ;
        if n <=m then
                summe := summe + 1:
        end if:
        summe ;
end proc:
A001055 := n -> T(n, n):
[seq(A001055(n), n=1..100)]; # Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)
  • Mathematica
    c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r] = Module[{ds, i}, ds = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson, Oct 28 2002 *)
T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] * T[n/#, #]&];
a[n_] := T[n, n];
a /@ Range[100] (* Jean-François Alcover, Jan 03 2020 *)
  • PARI
    /* factorizations of n with factors <= m (n,m positive integers) */
fcnt(n,m) = {local(s);s=0;if(n == 1,s=1,fordiv(n,d,if(d > 1 & d <= m,s=s+fcnt(n/d,d))));s}
A001055(n) = fcnt(n,n) \\ Michael B. Porter, Oct 29 2009
  • PARI
    \\ code using Dirichlet g.f., based on Somos's code for A007896
{a(n) = my(A, v, w, m);
if(
n<1, 0,
\\ define unit vector v = [1, 0, 0, ...] of length n
v = vector(n, k, k==1);
for(k=2, n,
m = #digits(n, k) - 1;
\\ expand 1/(1-x)^k out far enough
A = (1 - x)^ -1 + x * O(x^m);
\\ w = zero vector of length n
w = vector(n);
\\ convert A to a vector
for(i=0, m, w[k^i] = polcoeff(A, i));
\\ build the answer
v = dirmul(v, w)
);
v[n]
)
};
\\ produce the sequence
vector(100,n,a(n)) \\ N. J. A. Sloane, May 26 2014
  • PARI
    v=vector(100, k, k==1); for(n=2, #v, v+=dirmul(v, vector(#v, k, (k>1) && n^valuation(k,n)==k)) ); v \\ Max Alekseyev, Jul 16 2014
  • Python
    from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n<=m else 0
    A=filter(lambda d: d<=m, divisors(n)[1:-1])
    s=sum(T(n//d, d) for d in A)
    return s + 1 if n<=m else s
def a(n): return T(n, n)
print([a(n) for n in range(1, 106)]) # Indranil Ghosh, Aug 19 2017

Formula

The asymptotic behavior of this sequence was studied by Canfield, Erdős & Pomerance and Luca, Mukhopadhyay, & Srinivas. - Jonathan Vos Post, Jul 07 2008
Dirichlet g.f.: Product_{k>=2} 1/(1 - 1/k^s).
If n = p^k for a prime p, a(n) = partitions(k) = A000041(k).
Since the sequence a(n) is the right diagonal of A066032, the given recursive formula for A066032 applies (see Maple program). - Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)
a(A002110(n)) = A000110(n).
a(p^k*q^k) = A002774(k) if p and q are distinct primes. - R. J. Mathar, Jun 06 2024
a(n) = A028422(n) + 1. - Gus Wiseman, Feb 07 2025

Extensions

Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, Jun 08 2009

A008284 Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010: (Start)
A000041(n+1) = 1 + Sum_{r=1..n} Sum_{k=1..min(r,n-r+1)} T(r,k).
T(n, n-k) is also the number of partitions of k in which the greatest part is at most n-k. (End)
From Richard R. Forberg, Dec 26 2014: (Start)
Elements of T(n, k) for n <= 2+3k, equal A000041(n-k) - A000070(n-2k-1), with the assumption A000070(n) = 0 for n < 0.
The diagonal T(2+2k, k), for k > 1 equals A007042, and the diagonal T(3+3k,k), for k >= 1, equals A104384. (End)
T(-n, k) is used as a definition for A380038, which can therefore be seen as an extension of this sequence for negative n. - Friedjof Tellkamp, Jan 18 2025

Examples

			The triangle T(n,k) begins:
   n\k 1  2  3  4  5  6  7  8  9 10 11 12 ...
   1:  1
   2:  1  1
   3:  1  1  1
   4:  1  2  1  1
   5:  1  2  2  1  1
   6:  1  3  3  2  1  1
   7:  1  3  4  3  2  1  1
   8:  1  4  5  5  3  2  1  1
   9:  1  4  7  6  5  3  2  1  1
  10:  1  5  8  9  7  5  3  2  1  1
  11:  1  5 10 11 10  7  5  3  2  1  1
  12:  1  6 12 15 13 11  7  5  3  2  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 03 2012; additional extension by _Bob Selcoe_, Jun 09 2013
T(7,3) = 4 because we have [3,3,1], [3,2,2], [3,2,1,1] and [3,1,1,1,1], each having greatest part 3; or [5,1,1], [4,2,1], [3,3,1] and [3,2,2] each having 3 parts.
* Example from formula above: T(10,4) = 9 because T(6,4) + T(6,3) + T(6,2) + T(6,1) = 2 + 3 + 3 + 1 = 9.
* P(n) = P(n-1) + DT(n-1). P(n) = unordered partitions of n. (A000041) DT(n-1) = sum of diagonals beginning at T(n-1,1).
Example P(11) = 56, P(10) = 42, sum DT(10) = 1 + 4 + 5 + 3 + 1 = 14. - _Bob Selcoe_, Jun 09 2013
From _Omar E. Pol_, Nov 19 2019: (Start)
Illustration of initial terms: T(n,k) is also the number of vertical line segments in the k-th column of the n-th diagram, which represents the partitions of n:
.
    1    1 1    1 1 1    1 2 1 1    1 2 2 1 1    1 3 3 2 1 1    1 3 4 3 2 1 1
.
   _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |   _| | | | | | |
        _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |   _ _| | | | | |
               _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |   _ _ _| | | | |
                        _ _|   |   _ _|   | |   _ _|   | | |   _ _|   | | | |
                        _ _ _ _|   _ _ _ _| |   _ _ _ _| | |   _ _ _ _| | | |
                                   _ _ _|   |   _ _ _|   | |   _ _ _|   | | |
                                   _ _ _ _ _|   _ _ _ _ _| |   _ _ _ _ _| | |
                                                _ _|   |   |   _ _|   |   | |
                                                _ _ _ _|   |   _ _ _ _|   | |
                                                _ _ _|     |   _ _ _|     | |
                                                _ _ _ _ _ _|   _ _ _ _ _ _| |
                                                               _ _ _|   |   |
                                                               _ _ _ _ _|   |
                                                               _ _ _ _|     |
                                                               _ _ _ _ _ _ _|
(End)

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
  • D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions, Addison-Wesley Professional, 2005, pp. 38, 45, 50 [From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010]
  • D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 400.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 294.

Links

Crossrefs

A000041 is row sums and diagonal.
Columns k=1-10 are: A057427, A004526, A069905, A026810, A026811, A026812, A026813, A026814, A026815, A026816.
Partial sums of rows gives A026820.
Cf. A038497, A038498, A039805-A039809, A060016, A126988, A380038.
Read from right to left gives A058398.
Subtriangle of A072233 without row n=0 and column m=0.
Cf. A007042, A104384 which are diagonals with slope -2, -3.

Programs

  • Haskell
    a008284 n k = a008284_tabl !! (n-1) !! (k-1)
a008284_row n = a008284_tabl !! (n-1)
a008284_tabl = [1] : f [[1]] where
   f xss = ys : f (ys : xss) where
     ys = (map sum $ zipWith take [1..] xss) ++ [1]
-- Reinhard Zumkeller, Sep 05 2014
  • Maple
    G:=-1+1/product(1-t*x^j,j=1..15): Gser:=simplify(series(G,x=0,17)): for n from 1 to 14 do P[n]:=coeff(Gser,x^n) od: for n from 1 to 14 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 12 2006
with(combstruct):for n from 0 to 18 do seq(count(Partition(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Mar 30 2009
T := proc(n,k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else T(n - 1, k - 1) + T(n - k, k) fi end: seq(print(seq(T(n, k), k=1..n)),n=1..14); # Peter Luschny, Jul 24 2011
  • Mathematica
    Column[Table[ IntegerPartitions[n, {k}] // Length, {n, 1, 20}, {k, 1, n}], Center] (* Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010 *)
(*Recurrence closely related to natural numbers and number of divisors of n*)
Clear[t]; nn = 14; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, n - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0];Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]][[1 ;; 96]] (* Mats Granvik, Jan 01 2015 *)
Table[SeriesCoefficient[1/QPochhammer[a q, q], {q, 0, n}, {a, 0, k}], {n, 1, 15}, {k, 1, n}] // Column (* Vladimir Reshetnikov, Nov 18 2016 *)
T[n_, k_] := T[n, k] = If[n>0 && k>0, T[n-1, k-1] + T[n-k, k], Boole[n==0 && k==0]]
Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Robert A. Russell, May 12 2018 after Knuth 7.2.1.4 (39) *)
  • PARI
    T(n,k)=#partitions(n-k,k)
for(n=1,9,for(k=1,n,print1(T(n,k)", "))) \\ Charles R Greathouse IV, Jan 04 2016
  • PARI
    A8284=[]; A008284(n,k)={for(n=#A8284+1,n,A8284=concat(A8284,[vector(n,k,if(2*k1,A8284[n-k][k]+A8284[n-1][k-1],1),numbpart(n-k)))]));if(k,A8284[n][k],A8284[n])} \\ Without 2nd argument, return row n. - M. F. Hasler, Sep 26 2017
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A008284_T(n,k):
    if k==n or k==1: return 1
    if k>n: return 0
    return A008284_T(n-1,k-1)+A008284_T(n-k,k) # Chai Wah Wu, Sep 21 2023
  • Sage
    from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (1..n)] for n in (1..12)] # Peter Luschny, Aug 01 2015

Formula

T(n, k) = Sum_{i=1..k} T(n-k, i), for 1 <= k <= n-1; T(n, n) = 1 for n >= 1.
Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k > n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. for k-th column: x^k/(Product_{j=1..k} (1-x^j)). - Wolfdieter Lang, Nov 29 2000
G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^(P_n(y)/n), where P_n(y) = Sum_{d|n} eulerphi(n/d)*y^d. - Paul D. Hanna, Jul 13 2004
If k >= n/2, T(n,k) = T(2(n-k),n-k) = A000041(n-k). - Franklin T. Adams-Watters, Jan 12 2006 [Relation included by Hans Loeblich, Apr 16 2019, relation extended by Evan Robinson, Jun 30 2021]
G.f.: G(t,x) = -1 + 1/Product_{j>=1} (1-t*x^j). - Emeric Deutsch, Feb 12 2006
A002865(n) = Sum_{k=2..floor((n+2)/2)} T(n-k+1,k-1). - Reinhard Zumkeller, Nov 04 2007
A000700(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - Jeremy L. Martin, Jul 06 2013
G.f.: -1 + e^(F(x,z)), where F(x,z) = Sum_{n >= 1} (x*z)^n/(n*(1 - z^n)) is a g.f. for A126988. - Peter Bala, Jan 13 2015
Also, T(n, n-k) = k for k = 1, 2, 3; n >= 2k. T(n, 2) = floor(n/2). T(n, 3) = round(n^2/12). - M. F. Hasler, Sep 26 2017
T(n,k) = [n>0 & k>0] * (T(n-1,k-1) + T(n-k,k)) + [n==0 & k==0]. - Robert A. Russell, May 12 2018 from Knuth 7.2.1.4 (39)
T(n, k) = Sum_{i=0..n-1} T(n-ik-1, k-1) for k >= 1; T(-n, k) = 0 for n > 0; T(n, 0) = [n==0]. - Joshua Swanson (writing for Juexian Li), May 24 2020

A000166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540, 413496759611120779881, 9510425471055777937262
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Euler (1809) not only gives the first ten or so terms of the sequence, he also proves both recurrences a(n) = (n-1)*(a(n-1) + a(n-2)) and a(n) = n*a(n-1) + (-1)^n.
a(n) is the permanent of the matrix with 0 on the diagonal and 1 elsewhere. - Yuval Dekel, Nov 01 2003
a(n) is the number of desarrangements of length n. A desarrangement of length n is a permutation p of {1,2,...,n} for which the smallest of all the ascents of p (taken to be n if there are no ascents) is even. Example: a(3) = 2 because we have 213 and 312 (smallest ascents at i = 2). See the J. Désarménien link and the Bona reference (p. 118). - Emeric Deutsch, Dec 28 2007
a(n) is the number of deco polyominoes of height n and having in the last column an even number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. - Emeric Deutsch, Dec 28 2007
Attributed to Nicholas Bernoulli in connection with a probability problem that he presented. See Problem #15, p. 494, in "History of Mathematics" by David M. Burton, 6th edition. - Mohammad K. Azarian, Feb 25 2008
a(n) is the number of permutations p of {1,2,...,n} with p(1)!=1 and having no right-to-left minima in consecutive positions. Example a(3) = 2 because we have 231 and 321. - Emeric Deutsch, Mar 12 2008
a(n) is the number of permutations p of {1,2,...,n} with p(n)! = n and having no left to right maxima in consecutive positions. Example a(3) = 2 because we have 312 and 321. - Emeric Deutsch, Mar 12 2008
Number of wedged (n-1)-spheres in the homotopy type of the Boolean complex of the complete graph K_n. - Bridget Tenner, Jun 04 2008
The only prime number in the sequence is 2. - Howard Berman (howard_berman(AT)hotmail.com), Nov 08 2008
From Emeric Deutsch, Apr 02 2009: (Start)
a(n) is the number of permutations of {1,2,...,n} having exactly one small ascent. A small ascent in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. (Example: a(3) = 2 because we have 312 and 231; see the Charalambides reference, pp. 176-180.) [See also David, Kendall and Barton, p. 263. - N. J. A. Sloane, Apr 11 2014]
a(n) is the number of permutations of {1,2,...,n} having exactly one small descent. A small descent in a permutation (p_1,p_2,...,p_n) is a position i such that p_i - p_{i+1} = 1. (Example: a(3)=2 because we have 132 and 213.) (End)
For n > 2, a(n) + a(n-1) = A000255(n-1); where A000255 = (1, 1, 3, 11, 53, ...). - Gary W. Adamson, Apr 16 2009
Connection to A002469 (game of mousetrap with n cards): A002469(n) = (n-2)*A000255(n-1) + A000166(n). (Cf. triangle A159610.) - Gary W. Adamson, Apr 17 2009
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) is the sum of the values of the largest fixed points of all non-derangements of length n-1. Example: a(4)=9 because the non-derangements of length 3 are 123, 132, 213, and 321, having largest fixed points 3, 1, 3, and 2, respectively.
a(n) is the number of non-derangements of length n+1 for which the difference between the largest and smallest fixed point is 2. Example: a(3) = 2 because we have 1'43'2 and 32'14'; a(4) = 9 because we have 1'23'54, 1'43'52, 1'53'24, 52'34'1, 52'14'3, 32'54'1, 213'45', 243'15', and 413'25' (the extreme fixed points are marked).
(End)
a(n), n >= 1, is also the number of unordered necklaces with n beads, labeled differently from 1 to n, where each necklace has >= 2 beads. This produces the M2 multinomial formula involving partitions without part 1 given below. Because M2(p) counts the permutations with cycle structure given by partition p, this formula gives the number of permutations without fixed points (no 1-cycles), i.e., the derangements, hence the subfactorials with their recurrence relation and inputs. Each necklace with no beads is assumed to contribute a factor 1 in the counting, hence a(0)=1. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 01 2010
From Emeric Deutsch, Sep 06 2010: (Start)
a(n) is the number of permutations of {1,2,...,n, n+1} starting with 1 and having no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 1324 and 1432.
a(n) is the number of permutations of {1,2,...,n} that do not start with 1 and have no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 213 and 321.
(End)
Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleave except on the leftmost path, there is no vertex of outdegree one on the leftmost path. - Wenjin Woan, May 23 2011
a(n) is the number of zeros in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 2 pure options. - Raimundas Vidunas, Jan 22 2014
Convolution of sequence A135799 with the sequence generated by 1+x^2/(2*x+1). - Thomas Baruchel, Jan 08 2016
The number of interior lattice points of the subpolytope of the n-dimensional permutohedron whose vertices correspond to permutations avoiding 132 and 312. - Robert Davis, Oct 05 2016
Consider n circles of different radii, where each circle is either put inside some bigger circle or contains a smaller circle inside it (no common points are allowed). Then a(n) gives the number of such combinations. - Anton Zakharov, Oct 12 2016
If we partition the permutations of [n+1] in A000240 according to their starting digit, we will get (n+1) equinumerous classes each of size a(n), i.e., A000240(n+1) = (n+1)*a(n), hence a(n) is the size of each class of permutations of [n+1] in A000240. For example, for n = 4 we have 45 = 5*9. - Enrique Navarrete, Jan 10 2017
Call d_n1 the permutations of [n] that have the substring n1 but no substring in {12,23,...,(n-1)n}. If we partition them according to their starting digit, we will get (n-1) equinumerous classes each of size A000166(n-2) (the class starting with the digit 1 is empty since we must have the substring n1). Hence d_n1 = (n-1)*A000166(n-2) and A000166(n-2) is the size of each nonempty class in d_n1. For example, d_71 = 6*44 = 264, so there are 264 permutations in d_71 distributed in 6 nonempty classes of size A000166(5) = 44. (To get permutations in d_n1 recursively from more basic ones see the link "Forbidden Patterns" below.) - Enrique Navarrete, Jan 15 2017
Also the number of maximum matchings and minimum edge covers in the n-crown graph. - Eric W. Weisstein, Jun 14 and Dec 24 2017
The sequence a(n) taken modulo a positive integer k is periodic with exact period dividing k when k is even and dividing 2*k when k is odd. This follows from the congruence a(n+k) = (-1)^k*a(n) (mod k) for all n and k, which in turn is easily proved by induction making use of the recurrence a(n) = n*a(n-1) + (-1)^n. - Peter Bala, Nov 21 2017
a(n) is the number of distinct possible solutions for a directed, no self loop containing graph (not necessarily connected) that has n vertices, and each vertex has an in- and out-degree of exactly 1. - Patrik Holopainen, Sep 18 2018
a(n) is the dimension of the kernel of the random-to-top and random-to-random shuffling operators over a collection of n objects (in a vector space of size n!), as noticed by M. Wachs and V. Reiner. See the Reiner, Saliola and Welker reference below. - Nadia Lafreniere, Jul 18 2019
a(n) is the number of distinct permutations for a Secret Santa gift exchange with n participants. - Patrik Holopainen, Dec 30 2019
a(2*n+1) is even. More generally, a(m*n+1) is divisible by m*n, which follows from a(n+1) = n*(a(n) + a(n-1)) = n*A000255(n-1) for n >= 1. a(2*n) is odd; in fact, a(2*n) == 1 (mod 8). Other divisibility properties include a(6*n) == 1 (mod 24), a(9*n+4) == a(9*n+7) == 0 (mod 9), a(10*n) == 1 (mod 40), a(11*n+5) == 0 (mod 11) and a(13*n+8 ) == 0 (mod 13). - Peter Bala, Apr 05 2022
Conjecture: a(n) with n > 2 is a perfect power only for n = 4 with a(4) = 3^2. This has been verified for n <= 1000. - Zhi-Wei Sun, Jan 09 2025

Examples

			a(2) = 1, a(3) = 2 and a(4) = 9 since the possibilities are {BA}, {BCA, CAB} and {BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA}. - _Henry Bottomley_, Jan 17 2001
The Boolean complex of the complete graph K_4 is homotopy equivalent to the wedge of 9 3-spheres.
Necklace problem for n = 6: partitions without part 1 and M2 numbers for n = 6: there are A002865(6) = 4 such partitions, namely (6), (2,4), (3^2) and (2^3) in A-St order with the M2 numbers 5!, 90, 40 and 15, respectively, adding up to 265 = a(6). This corresponds to 1 necklace with 6 beads, two necklaces with 2 and 4 beads respectively, two necklaces with 3 beads each and three necklaces with 2 beads each. - _Wolfdieter Lang_, Jun 01 2010
G.f. = 1 + x^2 + 9*x^3 + 44*x^4 + 265*x^5 + 1854*x^6 + 14833*x^7 + 133496*x^8 + ...

References

  • U. Abel, Some new identities for derangement numbers, Fib. Q., 56:4 (2018), 313-318.
  • M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 32.
  • R. A. Brualdi and H. J. Ryser: Combinatorial Matrix Theory, 1992, Section 7.2, p. 202.
  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 182.
  • Florence Nightingale David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 168.
  • Florence Nightingale David, Maurice George Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263, Table 7.5.1, row 1.
  • P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
  • J. M. de Saint-Martin, "Le problème des rencontres" in Quadrature, No. 61, pp. 14-19, 2006, EDP-Sciences Les Ulis (France).
  • H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 19.
  • Leonhard Euler, Solution quaestionis curiosae ex doctrina combinationum, Mémoires Académie sciences St. Pétersburg 3 (1809/1810), 57-64; also E738 in his Collected Works, series I, volume 7, pages 435-440.
  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
  • A. Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
  • Irving Kaplansky, John Riordan, The problème des ménages. Scripta Math. 12 (1946), 113-124. See Eq(1).
  • Arnold Kaufmann, "Introduction à la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92.
  • Florian Kerschbaum and Orestis Terzidis, Filtering for Private Collaborative Benchmarking, in Emerging Trends in Information and Communication Security, Lecture Notes in Computer Science, Volume 3995/2006.
  • E. Lozansky and C. Rousseau, Winning Solutions, Springer, 1996; see p. 152.
  • P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 102.
  • M. S. Petković, "Non-attacking rooks", Famous Puzzles of Great Mathematicians, pp. 265-268, Amer. Math. Soc.(AMS), 2009.
  • V. Reiner, F. Saliola, and V. Welker. Spectra of Symmetrized Shuffling Operators, Memoirs of the American Mathematical Society, vol. 228, Amer. Math. Soc., Providence, RI, 2014, pp. 1-121. See section VI.9.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
  • H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 23.
  • T. Simpson, Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 122.
  • D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 82.
  • H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=1).

Links

Crossrefs

Cf. A000142, A002467, A003221, A000522, A000240, A000387, A000449, A000475, A129135, A092582, A000255, A002469, A159610, A068985, A068996, A047865, A038205, A008279, A281682.
For the probabilities a(n)/n!, see A053557/A053556 and A103816/A053556.
A diagonal of A008291 and A068106. Column A008290(n,0).
A001120 has a similar recurrence.
For other derangement numbers see also A053871, A033030, A088991, A088992.
Pairwise sums of A002741 and A000757. Differences of A001277.
Cf. A101560, A101559, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.
A diagonal in triangles A008305 and A010027.
a(n)/n! = A053557/A053556 = (N(n, n) of A103361)/(D(n, n) of A103360).
Column k=0 of A086764 and of A334715. Column k=1 of A364068.
Row sums of A216963 and of A323671.

Programs

  • Haskell
    a000166 n = a000166_list !! n
a000166_list = 1 : 0 : zipWith (*) [1..]
                       (zipWith (+) a000166_list $ tail a000166_list)
-- Reinhard Zumkeller, Dec 09 2012
  • Magma
    I:=[0,1]; [1] cat [n le 2 select I[n] else (n-1)*(Self(n-1)+Self(n-2)): n in [1..30]]; // Vincenzo Librandi, Jan 07 2016
  • Maple
    A000166 := proc(n) option remember; if n<=1 then 1-n else (n-1)*(procname(n-1)+procname(n-2)); fi; end;
a:=n->n!*sum((-1)^k/k!, k=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, May 17 2007
ZL1:=[S,{S=Set(Cycle(Z,card>1))},labeled]: seq(count(ZL1,size=n),n=0..21); # Zerinvary Lajos, Sep 26 2007
with (combstruct):a:=proc(m) [ZL,{ZL=Set(Cycle(Z,card>=m))},labeled]; end: A000166:=a(2):seq(count(A000166,size=n),n=0..21); # Zerinvary Lajos, Oct 02 2007
Z := (x, m)->m!^2*sum(x^j/((m-j)!^2), j=0..m): R := (x, n, m)->Z(x, m)^n: f := (t, n, m)->sum(coeff(R(x, n, m), x, j)*(t-1)^j*(n*m-j)!, j=0..n*m): seq(f(0, n, 1), n=0..21); # Zerinvary Lajos, Jan 22 2008
a:=proc(n) if `mod`(n,2)=1 then sum(2*k*factorial(n)/factorial(2*k+1), k=1.. floor((1/2)*n)) else 1+sum(2*k*factorial(n)/factorial(2*k+1), k=1..floor((1/2)*n)-1) end if end proc: seq(a(n),n=0..20); # Emeric Deutsch, Feb 23 2008
G(x):=2*exp(-x)/(1-x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/2,n=0..21); # Zerinvary Lajos, Apr 03 2009
seq(simplify(KummerU(-n, -n, -1)), n = 0..23); # Peter Luschny, May 10 2022
  • Mathematica
    a[0] = 1; a[n_] := n*a[n - 1] + (-1)^n; a /@ Range[0, 21] (* Robert G. Wilson v *)
a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 (* Michael Taktikos, May 26 2006 *)
Range[0, 20]! CoefficientList[ Series[ Exp[ -x]/(1 - x), {x, 0, 20}], x]
dr[{n_,a1_,a2_}]:={n+1,a2,n(a1+a2)}; Transpose[NestList[dr,{0,0,1},30]][[3]] (* Harvey P. Dale, Feb 23 2013 *)
a[n_] := (-1)^n HypergeometricPFQ[{- n, 1}, {}, 1]; (* Michael Somos, Jun 01 2013 *)
a[n_] := n! SeriesCoefficient[Exp[-x] /(1 - x), {x, 0, n}]; (* Michael Somos, Jun 01 2013 *)
Table[Subfactorial[n], {n, 0, 21}] (* Jean-François Alcover, Jan 10 2014 *)
RecurrenceTable[{a[n] == n*a[n - 1] + (-1)^n, a[0] == 1}, a, {n, 0, 23}] (* Ray Chandler, Jul 30 2015 *)
Subfactorial[Range[0, 20]] (* Eric W. Weisstein, Dec 31 2017 *)
nxt[{n_,a_}]:={n+1,a(n+1)+(-1)^(n+1)}; NestList[nxt,{0,1},25][[All,2]] (* Harvey P. Dale, Jun 01 2019 *)
  • Maxima
    s[0]:1$
s[n]:=n*s[n-1]+(-1)^n$
makelist(s[n],n,0,12); /* Emanuele Munarini, Mar 01 2011 */
  • PARI
    {a(n) = if( n<1, 1, n * a(n-1) + (-1)^n)}; /* Michael Somos, Mar 24 2003 */
  • PARI
    {a(n) = n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}; /* Michael Somos, Mar 24 2003 */
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m^m*x^m/(1+(m+1)*x+x*O(x^n))^(m+1)),n)} /* Paul D. Hanna */
  • PARI
    A000166=n->n!*sum(k=0,n,(-1)^k/k!) \\ M. F. Hasler, Jan 26 2012
  • PARI
    a(n)=if(n,round(n!/exp(1)),1) \\ Charles R Greathouse IV, Jun 17 2012
  • PARI
    apply( {A000166(n)=n!\/exp(n>0)}, [0..22]) \\ M. F. Hasler, Nov 09 2024
  • Python
    See Hobson link.
  • Python
    A000166_list, m, x = [], 1, 1
for n in range(10*2):
    x, m = x*n + m, -m
    A000166_list.append(x) # Chai Wah Wu, Nov 03 2014

Formula

(A000166 + A000522)/2 = A009179, (A000166 - A000522)/2 = A009628.
a(n) = A008290(n,0).
a(n) + A003048(n+1) = 2*n!. - D. G. Rogers, Aug 26 2006
a(n) = {(n-1)!/exp(1)}, n > 1, where {x} is the nearest integer function. - Simon Plouffe, March 1993 [This uses offset 1, see below for the version with offset 0. - Charles R Greathouse IV, Jan 25 2012]
a(0) = 1, a(n) = round(n!/e) = floor(n!/e + 1/2) for n > 0.
a(n) = n!*Sum_{k=0..n} (-1)^k/k!.
D-finite with recurrence a(n) = (n-1)*(a(n-1) + a(n-2)), n > 0.
a(n) = n*a(n-1) + (-1)^n.
E.g.f.: exp(-x)/(1-x).
a(n) = Sum_{k=0..n} binomial(n, k)*(-1)^(n-k)*k! = Sum_{k=0..n} (-1)^(n-k)*n!/(n-k)!. - Paul Barry, Aug 26 2004
The e.g.f. y(x) satisfies y' = x*y/(1-x).
Inverse binomial transform of A000142. - Ross La Haye, Sep 21 2004
In Maple notation, representation as n-th moment of a positive function on [-1, infinity]: a(n)= int( x^n*exp(-x-1), x=-1..infinity ), n=0, 1... . a(n) is the Hamburger moment of the function exp(-1-x)*Heaviside(x+1). - Karol A. Penson, Jan 21 2005
a(n) = A001120(n) - n!. - Philippe Deléham, Sep 04 2005
a(n) = Integral_{x=0..oo} (x-1)^n*exp(-x) dx. - Gerald McGarvey, Oct 14 2006
a(n) = Sum_{k=2,4,...} T(n,k), where T(n,k) = A092582(n,k) = k*n!/(k+1)! for 1 <= k < n and T(n,n)=1. - Emeric Deutsch, Feb 23 2008
a(n) = n!/e + (-1)^n*(1/(n+2 - 1/(n+3 - 2/(n+4 - 3/(n+5 - ...))))). Asymptotic result (Ramanujan): (-1)^n*(a(n) - n!/e) ~ 1/n - 2/n^2 + 5/n^3 - 15/n^4 + ..., where the sequence [1,2,5,15,...] is the sequence of Bell numbers A000110. - Peter Bala, Jul 14 2008
From William Vaughn (wvaughn(AT)cvs.rochester.edu), Apr 13 2009: (Start)
a(n) = Integral_{p=0..1} (log(1/(1-p)) - 1)^n dp.
Proof: Using the substitutions 1=log(e) and y = e(1-p) the above integral can be converted to ((-1)^n/e) Integral_{y=0..e} (log(y))^n dy.
From CRC Integral tables we find the antiderivative of (log(y))^n is (-1)^n n! Sum_{k=0..n} (-1)^k y(log(y))^k / k!.
Using the fact that e(log(e))^r = e for any r >= 0 and 0(log(0))^r = 0 for any r >= 0 the integral becomes n! * Sum_{k=0..n} (-1)^k / k!, which is line 9 of the Formula section. (End)
a(n) = exp(-1)*Gamma(n+1,-1) (incomplete Gamma function). - Mark van Hoeij, Nov 11 2009
G.f.: 1/(1-x^2/(1-2x-4x^2/(1-4x-9x^2/(1-6x-16x^2/(1-8x-25x^2/(1-... (continued fraction). - Paul Barry, Nov 27 2009
a(n) = Sum_{p in Pano1(n)} M2(p), n >= 1, with Pano1(n) the set of partitions without part 1, and the multinomial M2 numbers. See the characteristic array for partitions without part 1 given by A145573 in Abramowitz-Stegun (A-S) order, with A002865(n) the total number of such partitions. The M2 numbers are given for each partition in A-St order by the array A036039. - Wolfdieter Lang, Jun 01 2010
a(n) = row sum of A008306(n), n > 1. - Gary Detlefs, Jul 14 2010
a(n) = ((-1)^n)*(n-1)*hypergeom([-n+2, 2], [], 1), n>=1; 1 for n=0. - Wolfdieter Lang, Aug 16 2010
a(n) = (-1)^n * hypergeom([ -n, 1], [], 1), n>=1; 1 for n=0. From the binomial convolution due to the e.g.f. - Wolfdieter Lang, Aug 26 2010
Integral_{x=0..1} x^n*exp(x) = (-1)^n*(a(n)*e - n!).
O.g.f.: Sum_{n>=0} n^n*x^n/(1 + (n+1)*x)^(n+1). - Paul D. Hanna, Oct 06 2011
Abs((a(n) + a(n-1))*e - (A000142(n) + A000142(n-1))) < 2/n. - Seiichi Kirikami, Oct 17 2011
G.f.: hypergeom([1,1],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
From Sergei N. Gladkovskii, Nov 25 2011, Jul 05 2012, Sep 23 2012, Oct 13 2012, Mar 09 2013, Mar 10 2013, Oct 18 2013: (Start)
Continued fractions:
In general, e.g.f. (1+a*x)/exp(b*x) = U(0) with U(k) = 1 + a*x/(1-b/(b-a*(k+1)/U(k+1))). For a=-1, b=-1: exp(-x)/(1-x) = 1/U(0).
E.g.f.: (1-x/(U(0)+x))/(1-x), where U(k) = k+1 - x + (k+1)*x/U(k+1).
E.g.f.: 1/Q(0) where Q(k) = 1 - x/(1 - 1/(1 - (k+1)/Q(k+1))).
G.f.: 1/U(0) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: Q(0)/(1+x) where Q(k) = 1 + (2*k+1)*x/((1+x)-2*x*(1+x)*(k+1)/(2*x*(k+1)+(1+x)/ Q(k+1))).
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2-(1-2*x*k)*(1-2*x-2*x*k)/T(k+1)). (End)
0 = a(n)*(a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n>=0. - Michael Somos, Jan 25 2014
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^k*(k + x + 1)^(n-k) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^(n-k)*(k + x - 1)^k, for arbitrary x. - Peter Bala, Feb 19 2017
From Peter Luschny, Jun 20 2017: (Start)
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(-j-1, -n-1)*abs(Stirling1(j, k)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Pochhammer(n-k+1, k) (cf. A008279). (End)
a(n) = n! - Sum_{j=0..n-1} binomial(n,j) * a(j). - Alois P. Heinz, Jan 23 2019
Sum_{n>=2} 1/a(n) = A281682. - Amiram Eldar, Nov 09 2020
a(n) = KummerU(-n, -n, -1). - Peter Luschny, May 10 2022
a(n) = (-1)^n*Sum_{k=0..n} Bell(k)*Stirling1(n+1, k+1). - Mélika Tebni, Jul 05 2022

Extensions

Minor edits by M. F. Hasler, Jan 16 2017

A006128 Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 275, 399, 556, 780, 1068, 1463, 1965, 2644, 3498, 4630, 6052, 7899, 10206, 13174, 16851, 21522, 27294, 34545, 43453, 54563, 68135, 84927, 105366, 130462, 160876, 198014, 242812, 297201, 362587, 441546, 536104, 649791, 785437, 947812, 1140945, 1371173, 1644136, 1968379, 2351597, 2805218, 3339869, 3970648, 4712040, 5584141, 6606438, 7805507, 9207637
Offset: 0

Views

Author

N. J. A. Sloane, Colin Mallows

Keywords

Comments

a(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989).) - Wolfdieter Lang
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g., for n=3 to n=2 we have a(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, see A066186. - Thomas Wieder, May 20 2004
Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel, Nov 07 2005
Starting with offset 1, = the partition triangle A008284 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008
Starting with offset 1 equals A000041: (1, 1, 2, 3, 5, 7, 11, ...) convolved with A000005: (1, 2, 2, 3, 2, 4, ...). - Gary W. Adamson, Jun 16 2009
Apart from initial 0 row sums of triangle A066633, also the Möbius transform is A085410. - Gary W. Adamson, Mar 21 2011
More generally, the total number of parts >= k in all partitions of n equals the sum of k-th largest parts of all partitions of n. In this case k = 1. Apart from initial 0 the first column of A181187. - Omar E. Pol, Feb 14 2012
Row sums of triangle A221530. - Omar E. Pol, Jan 21 2013
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the total number of divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also as follows:
Convolution of A000005 and A000041.
Convolution of A006218 and A002865.
Convolution of A341062 and A000070.
Row sums of triangles A221531, A245095, A339258, A340525, A340529. (End)
Number of ways to choose a part index of an integer partition of n, i.e., partitions of n with a selected position. Selecting a part value instead of index gives A000070. - Gus Wiseman, Apr 19 2021

Examples

			For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. The total number of parts is 12. On the other hand, the sum of the largest parts of all partitions is 4 + 2 + 3 + 2 + 1 = 12, equaling the total number of parts, so a(4) = 12. - _Omar E. Pol_, Oct 12 2018

References

  • S. M. Luthra, On the average number of summands in partitions of n, Proc. Nat. Inst. Sci. India Part. A, 23 (1957), p. 483-498.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A093694, A008284, A000005, A066633, A085410, A046746, A209423, A285220, A336902, A336903.
Main diagonal of A210485.
Column k=1 of A256193.
The version for normal multisets is A001787.
The unordered version is A001792.
The strict case is A015723.
The version for factorizations is A066637.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A336875 counts compositions with a selected part.
A339564 counts factorizations with a selected factor.
Cf. A066186, A083710, A130689, A264401, A276024, A343341.

Programs

  • GAP
    List([0..60],n->Length(Flat(Partitions(n)))); # Muniru A Asiru, Oct 12 2018
  • Haskell
    a006128 = length . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013
  • Maple
    g:= add(n*x^n*mul(1/(1-x^k), k=1..n), n=1..61):
a:= n-> coeff(series(g,x,62),x,n):
seq(a(n), n=0..61);
# second Maple program:
a:= n-> add(combinat[numbpart](n-j)*numtheory[tau](j), j=1..n):
seq(a(n), n=0..61);  # Alois P. Heinz, Aug 23 2019
  • Mathematica
    a[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; Table[ a[n], {n, 0, 41}]
CoefficientList[ Series[ Sum[n*x^n*Product[1/(1 - x^k), {k, n}], {n, 100}], {x, 0, 100}], x]
a[n_] := Plus @@ Max /@ IntegerPartitions@ n; Array[a, 45] (* Robert G. Wilson v, Apr 12 2011 *)
Join[{0}, ((Log[1 - x] + QPolyGamma[1, x])/(Log[x] QPochhammer[x]) + O[x]^60)[[3]]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Length /@ Table[IntegerPartitions[n] // Flatten, {n, 50}] (* Shouvik Datta, Sep 12 2021 *)
  • PARI
    f(n)= {local(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]1,i--;s+=i*(v[i]=(n-s)\i));t+=sum(k=1,n,v[k]));t } /* Thomas Baruchel, Nov 07 2005 */
  • PARI
    a(n) = sum(m=1, n, numdiv(m)*numbpart(n-m)) \\ Michel Marcus, Jul 13 2013
  • Python
    from sympy import divisor_count, npartitions
def a(n): return sum([divisor_count(m)*npartitions(n - m) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 25 2017

Formula

G.f.: Sum_{n>=1} n*x^n / Product_{k=1..n} (1-x^k).
G.f.: Sum_{k>=1} x^k/(1-x^k) / Product_{m>=1} (1-x^m).
a(n) = Sum_{k=1..n} k*A008284(n, k).
a(n) = Sum_{m=1..n} of the number of divisors of m * number of partitions of n-m.
Note that the formula for the above comment is a(n) = Sum_{m=1..n} d(m)*p(n-m) = Sum_{m=1..n} A000005(m)*A000041(n-m), if n >= 1. - Omar E. Pol, Jan 21 2013
Erdős and Lehner show that if u(n) denotes the average largest part in a partition of n, then u(n) ~ constant*sqrt(n)*log n.
a(n) = A066897(n) + A066898(n), n>0. - Reinhard Zumkeller, Mar 09 2012
a(n) = A066186(n) - A196087(n), n >= 1. - Omar E. Pol, Apr 22 2012
a(n) = A194452(n) + A024786(n+1). - Omar E. Pol, May 19 2012
a(n) = A000203(n) + A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{m=1..p(n)} A194446(m) = Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1. - Omar E. Pol, May 12 2013
a(n) = A198381(n) + A026905(n), n >= 1. - Omar E. Pol, Aug 10 2013
a(n) = O(sqrt(n)*log(n)*p(n)), where p(n) is the partition function A000041(n). - Peter Bala, Dec 23 2013
a(n) = Sum_{m=1..n} A006218(m)*A002865(n-m), n >= 1. - Omar E. Pol, Jul 14 2014
From Vaclav Kotesovec, Jun 23 2015: (Start)
Asymptotics (Luthra, 1957): a(n) = p(n) * (C*N^(1/2) + C^2/2) * (log(C*N^(1/2)) + gamma) + (1+C^2)/4 + O(N^(-1/2)*log(N)), where N = n - 1/24, C = sqrt(6)/Pi, gamma is the Euler-Mascheroni constant A001620 and p(n) is the partition function A000041(n).
The formula a(n) = p(n) * (sqrt(3*n/(2*Pi)) * (log(n) + 2*gamma - log(Pi/6)) + O(log(n)^3)) in the abstract of the article by Kessler and Livingston (cited also in the book by Sandor, p. 495) is incorrect!
Right is: a(n) = p(n) * (sqrt(3*n/2)/Pi * (log(n) + 2*gamma - log(Pi^2/6)) + O(log(n)^3))
or a(n) ~ exp(Pi*sqrt(2*n/3)) * (log(6*n/Pi^2) + 2*gamma) / (4*Pi*sqrt(2*n)).
(End)
G.f.: (log(1-x) + psi_x(1))/(log(x) * (x)inf), where psi_q(z) is the q-digamma function, and (q)_inf is the q-Pochhammer symbol (the Euler function). - _Vladimir Reshetnikov, Nov 17 2016
a(n) = Sum_{m=1..n} A341062(m)*A000070(n-m), n >= 1. - Omar E. Pol, Feb 05 2021 2014

A066186 Sum of all parts of all partitions of n.

Original entry on oeis.org

0, 1, 4, 9, 20, 35, 66, 105, 176, 270, 420, 616, 924, 1313, 1890, 2640, 3696, 5049, 6930, 9310, 12540, 16632, 22044, 28865, 37800, 48950, 63336, 81270, 104104, 132385, 168120, 212102, 267168, 334719, 418540, 520905, 647172, 800569, 988570, 1216215, 1493520
Offset: 0

Views

Author

Wouter Meeussen, Dec 15 2001

Keywords

Comments

Sum of the zeroth moments of all partitions of n.
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that any part z is composed of labeled elements of amount 1, i.e., z = 1_1 + 1_2 + ... + 1_z. Then one can take from z a single element in z different ways. E.g., for n=3 to n=2 we have A066186(3) = 9 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [111], [12] --> [111], [12] --> [2], [3] --> 2, [3] --> 2, [3] --> 2. For the unlabeled case, one can take a single element from z in only one way. Then the number of one-element transitions from the integer partitions of n to the partitions of n-1 is given by A000070. E.g., A000070(3) = 4 and for the transition from n=3 to n=2 one has [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. - Thomas Wieder, May 20 2004
Also sum of all parts of all regions of n (Cf. A206437). - Omar E. Pol, Jan 13 2013
From Omar E. Pol, Jan 19 2021: (Start)
Apart from initial zero this is also as follows:
Convolution of A000203 and A000041.
Convolution of A024916 and A002865.
For n >= 1, a(n) is also the number of cells in a symmetric polycube in which the terraces are the symmetric representation of sigma(k), for k = n..1, (cf. A237593) starting from the base and located at the levels A000041(0)..A000041(n-1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a three-dimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also the convolution of A340793 and A000070. (End)

Examples

			a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.
a(4)=20 because A000041(4)=5 and 4*5=20.

Links

Crossrefs

Cf. A000041, A093694, A000070, A132825, A001787 (same for ordered partitions), A277029, A000203, A221529, A237593, A239660.
First differences give A138879. - Omar E. Pol, Aug 16 2013
Row sums of triangles A138785, A181187, A245099, A337209, A339106, A340423, A340424, A221529, A302246, A338156, A340035, A340056, A340057, A346741. - Omar E. Pol, Aug 02 2021

Programs

  • Haskell
    a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013
  • Maple
    with(combinat): a:= n-> n*numbpart(n): seq(a(n), n=0..50); # Zerinvary Lajos, Apr 25 2007
  • Mathematica
    PartitionsP[ Range[0, 60] ] * Range[0, 60]
  • PARI
    a(n)=numbpart(n)*n \\ Charles R Greathouse IV, Mar 10 2012
  • Python
    from sympy import npartitions
def A066186(n): return n*npartitions(n) # Chai Wah Wu, Oct 22 2023
  • Sage
    [n*Partitions(n).cardinality() for n in range(41)] # Peter Luschny, Jul 29 2014

Formula

a(n) = n * A000041(n). - Omar E. Pol, Oct 10 2011
G.f.: x * (d/dx) Product_{k>=1} 1/(1-x^k), i.e., derivative of g.f. for A000041. - Jon Perry, Mar 17 2004 (adjusted to match the offset by Geoffrey Critzer, Nov 29 2014)
Equals A132825 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = A066967(n) + A066966(n). - Omar E. Pol, Mar 10 2012
a(n) = A207381(n) + A207382(n). - Omar E. Pol, Mar 13 2012
a(n) = A006128(n) + A196087(n). - Omar E. Pol, Apr 22 2012
a(n) = A220909(n)/2. - Omar E. Pol, Jan 13 2013
a(n) = Sum_{k=1..n} A000203(k)*A000041(n-k), n >= 1. - Omar E. Pol, Jan 20 2013
a(n) = Sum_{k=1..n} k*A036043(n,n-k+1). - L. Edson Jeffery, Aug 03 2013
a(n) = Sum_{k=1..n} A024916(k)*A002865(n-k), n >= 1. - Omar E. Pol, Jul 13 2014
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016
a(n) = Sum_{k=1..n} A340793(k)*A000070(n-k), n >= 1. - Omar E. Pol, Feb 04 2021

Extensions

a(0) added by Franklin T. Adams-Watters, Jul 28 2014

A025147 Number of partitions of n into distinct parts >= 2.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 17, 21, 25, 29, 35, 41, 48, 56, 66, 76, 89, 103, 119, 137, 159, 181, 209, 239, 273, 312, 356, 404, 460, 522, 591, 669, 757, 853, 963, 1085, 1219, 1371, 1539, 1725, 1933, 2164, 2418, 2702, 3016, 3362, 3746, 4171, 4637, 5155
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

From R. J. Mathar, Jul 31 2008: (Start)
These "partitions of n into distinct parts >= k" and "partitions of n into distinct parts, the least being k-1" come in pairs of similar, almost shifted but not identical, sequences:
A025147, A096765 (k=2)
A025148, A096749 (k=3)
A025149, A026824 (k=4)
A025150, A026825 (k=5)
A025151, A026826 (k=6)
A025152, A026827 (k=7)
A025153, A026828 (k=8)
A025154, A026829 (k=9)
A025155, A026830 (k=10)
A096740, A026831 (k=11)
The distinction in the definitions is that "distinct parts >= k" sets a lower bound to all parts, whereas "the least being ..." means that the lower limit must be attained by one of the parts. (End)
From N. J. A. Sloane, Sep 28 2008: (Start)
Generating functions and Maple programs for the sequences in the first and second columns of the above list are respectively:
For A025147, A025148, etc.:
f:=proc(k) product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end;
For A096765, A096749, etc.:
g:=proc(k) x^(k-1)*product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end; (End)
Also number of partitions of n+1 into distinct parts, the least being 1.
Number of different sums from 1+[1,3]+[1,4]+...+[1,n]. - Jon Perry, Jan 01 2004
Also number of partitions of n such that if k is the largest part, then all parts from 1 to k occur, k occurring at least twice. Example: a(7)=3 because we have [2,2,2,1],[2,2,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 09 2006
Also number of partitions of n+1 such that if k is the largest part, then all parts from 1 to k occur, k occurring exactly once. Example: a(7)=3 because we have [3,2,2,1],[3,2,1,1,1] and [2,1,1,1,1,1,1] (there is a simple bijection with the partitions defined before). - Emeric Deutsch, Apr 09 2006
Also number of partitions of n+1 into distinct parts where the number of parts is itself a part. - Reinhard Zumkeller, Nov 04 2007
Partial sums give A038348 (observed by Jonathan Vos Post, proved by several correspondents).
Trivially, number of partitions of n into distinct parts (as ascending lists) such that the first part is not 1, the second not 2, the third not 3, etc., see example. - Joerg Arndt, Jun 10 2013
Convolution with A033999 gives A270144 (apart from the offset). - R. J. Mathar, Jun 18 2016

Examples

			a(7) = 3, from {{3, 4}, {2, 5}, {7}}
From _Joerg Arndt_, Jun 10 2013: (Start)
There are a(17) = 21 partitions of 17 into distinct parts >=2:
01:  [ 2 3 4 8 ]
02:  [ 2 3 5 7 ]
03:  [ 2 3 12 ]
04:  [ 2 4 5 6 ]
05:  [ 2 4 11 ]
06:  [ 2 5 10 ]
07:  [ 2 6 9 ]
08:  [ 2 7 8 ]
09:  [ 2 15 ]
10:  [ 3 4 10 ]
11:  [ 3 5 9 ]
12:  [ 3 6 8 ]
13:  [ 3 14 ]
14:  [ 4 5 8 ]
15:  [ 4 6 7 ]
16:  [ 4 13 ]
17:  [ 5 12 ]
18:  [ 6 11 ]
19:  [ 7 10 ]
20:  [ 8 9 ]
21:  [ 17 ]
(End)

References

  • Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
  • Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Links

Crossrefs

Cf. A015744, A015745, A015746, A015750, A015753, A015754, A015755, A002865.

Programs

  • Haskell
    a025147 = p 2 where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k + 1) (m - k) + p (k + 1) m
-- Reinhard Zumkeller, Dec 28 2011
  • Maple
    g:=product(1+x^j,j=2..65): gser:=series(g,x=0,62): seq(coeff(gser,x,n),n=0..57); # Emeric Deutsch, Apr 09 2006
with(combstruct):ZL := {L = PowerSet(Sequence(Z,card>=2)) },unlabeled:seq(count([L,ZL],size=i),i=0..57); # Zerinvary Lajos, Mar 09 2007
  • Mathematica
    CoefficientList[Series[Product[1+q^n, {n, 2, 60}], {q, 0, 60}], q]
FoldList[ PartitionsQ[ #2+1 ]-#1&, 0, Range[ 64 ] ]
(* also *)
d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 2 &]; Table[d[n], {n, 12}] (* strict partitions, parts >= 2 *)
Table[Length[d[n]], {n, 40}] (* A025147 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
p[, 0] = 1; p[k, m_] := p[k, m] = If[m < k, 0, p[k+1, m-k] + p[k+1, m]]; Table[p[2, m], {m, 0, 59}] (* Jean-François Alcover, Apr 17 2014, after Reinhard Zumkeller *)
  • PARI
    a(n)=if(n,my(v=partitions(n));sum(i=1,#v,v[i][1]>1&&v[i]==vecsort(v[i],,8)),1) \\ Charles R Greathouse IV, Nov 20 2012

Formula

G.f.: Product_{k>=2} (1+x^k).
a(n) = A000009(n)-a(n-1) = Sum_{0<=k<=n} (-1)^k*A000009(n-k). - Henry Bottomley, May 09 2002
a(n)=t(n, 1), where t(n, k)=1+Sum_{i>j>k and i+j=n} t(i, j), 2<=k<=n. - Reinhard Zumkeller, Jan 01 2003
G.f.: 1 + Sum_{k=1..infinity} (x^(k*(k+3)/2) / Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 09 2006
The previous g.f. is a special case of the g.f. for partitions into distinct parts >= L, Sum_{n>=0} ( x^(n*(n+2*L-1)/2) / Product_{k=1..n} (1-x^k) ). - Joerg Arndt, Mar 24 2011
G.f.: Sum_{n>=1} ( x^(n*(n+1)/2-1) / Product_{k=1..n-1} (1-x^k) ), a special case of the g.f. for partitions into distinct parts >= L, Sum_{n>=L-1} ( x^(n*(n+1)/2-L*(L-1)/2) / Product_{k=1..n-(L-1)} (1-x^k) ). - Joerg Arndt, Mar 27 2011
a(n) = Sum_{1A060016(n-k+1,k-1), for n>0. - Reinhard Zumkeller, Nov 04 2007
a(n) = A096765(n+1). - R. J. Mathar, Jul 31 2008
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ 1/2 * A000009(n).
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)).
(End)

Extensions

Corrected and extended by Dean Hickerson, Oct 10 2001

A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 5, 1, 1, 0, 1, 7, 4, 2, 1, 0, 1, 11, 3, 2, 1, 1, 0, 1, 15, 8, 3, 3, 1, 1, 0, 1, 22, 7, 6, 2, 2, 1, 1, 0, 1, 30, 15, 6, 5, 3, 2, 1, 1, 0, 1, 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1, 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2010

Keywords

Comments

For the definition of "section" of the set of partitions of n see A135010.
Also, column 1 gives the number of partitions of n-1. For k >= 2, row n lists the number of k's in all partitions of n that do not contain 1 as a part.
From Omar E. Pol, Feb 12 2012: (Start)
It appears that reversed rows converge to A002865.
It appears that row n is also the base of an isosceles triangle in which the column sums give the partition numbers A000041 in descending order starting with p(n-1) = A000041(n-1). Example for n = 7:
.
. 1,
. 1, 0, 1,
. 4, 2, 1, 0, 1,
11, 3, 2, 1, 1, 0, 1,
---------------------
11, 7, 5, 3, 2, 1, 1,
.
It appears that in row n starts an infinite trapezoid in which column sums always give the number of partitions of n-1. Example for n = 7:
.
11, 3, 2, 1, 1, 0, 1,
. 8, 3, 3, 1, 1, 0, 1,
. 6, 2, 2, 1, 1, 0, 1,
. 5, 3, 2, 1, 1, 0, 1,
. 4, 2, 2, 1, 1, 0, 1,
. 5, 2, 2, 1, 1, 0,...
. 4, 2, 2, 1, 1,...
. 4, 2, 2, 1,...
. 4, 2, 2,...
. 4, 2,...
. 4,...
.
The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.
It appears that the first term of row n is one of the vertices of an infinite isosceles triangle in which column sums give the partition numbers A000041 in ascending order starting with p(n-1) = A000041(n-1). Example for n = 7:
11,
. 8,
. 7, 6,
. 6, 5,
. 10, 5, ...
. 10, ...
. 10, ...
-------------------
11, 15, 22, 30, ...
(End)
It appears that row n lists the first differences of the row n of triangle A207031 together with 1 (as the final term of row n). - Omar E. Pol, Feb 26 2012
More generally T(n,k) is the number of occurrences of k in the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Oct 21 2013

Examples

			Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031.
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of this triangle.
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So, for k = 1..7, row 7 gives: 11, 3, 2, 1, 1, 0, 1.
Triangle begins:
   1;
   1,  1;
   2,  0,  1;
   3,  2,  0,  1;
   5,  1,  1,  0, 1;
   7,  4,  2,  1, 0, 1;
  11,  3,  2,  1, 1, 0, 1;
  15,  8,  3,  3, 1, 1, 0, 1;
  22,  7,  6,  2, 2, 1, 1, 0, 1;
  30, 15,  6,  5, 3, 2, 1, 1, 0, 1;
  42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1;
  56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
  ...

Links

Crossrefs

Row sums give A138137. Where records occur is A134869.
Columns 1-10: A000041, A182712-A182714, A206555-A206560.
Sub-triangles (1-11): A023531, A129186, A194702-A194710
Cf. A066633, A135010, A182742, A182743, A194812, A206563, A207031, A207032, A206437, A211025.

Programs

  • Maple
    p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n,i) option remember; local g;
      if n=0        then [1]
    elif n<2 or i<2 then [0]
    else g:=   `if`(i>n, [0],  b(n-i, i));
         p(p([0$j=2..i, g[1]], b(n, i-1)), g)
      fi
    end:
h:= proc(n) option remember;
      `if`(n=0, 1, b(n, n)[1]+h(n-1))
    end:
T:= proc(n) h(n-1), b(n, n)[2..n][] end:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 19 2012
  • Mathematica
    p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1}, n<2 || i<2, {0}, True, g = If [i>n, {0}, b[n-i, i]]; p[p[Append[Array[0&, i-1], g[[1]]], b[n, i-1]], g]]]; h[n_] := h[n] = If[n == 0, 1, b[n, n][[1]] + h[n-1]]; t[n_] := {h[n-1], Sequence @@ b[n, n][[2 ;; n]]}; Table[t[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)
Table[{PartitionsP[n-1]}~Join~Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], k], {k,2,n}], {n,1,12}]  // Flatten (* Robert Price, May 15 2020 *)

Formula

It appears that T(n,k) = A207032(n,k) - A207032(n,k+2). - Omar E. Pol, Feb 26 2012

A301987 Heinz numbers of integer partitions whose product is equal to their sum.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 200, 211, 223, 227, 229, 233, 239, 241, 251
Offset: 1

Views

Author

Gus Wiseman, Mar 30 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			Sequence of reversed integer partitions begins: (1), (2), (3), (4), (2 2), (5), (6), (7), (8), (9), (10), (1 2 3), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (1 1 2 4), (24), (25), (26), (27), (28), (1 1 2 2 2), (29), (30).

Links

Crossrefs

Cf. A001055, A002865, A003963, A056239, A276024, A284640, A296150, A299701, A301957, A301988.

Programs

  • Maple
    q:= n-> (l-> mul(i, i=l)=add(i, i=l))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..300])[];  # Alois P. Heinz, Mar 27 2019
  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[300],Total[primeMS[#]]===Times@@primeMS[#]&]
Previous Showing 21-30 of 410 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.