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.

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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

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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

A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.

Original entry on oeis.org

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589, 801, 1048, 1407, 1820, 2399, 3087, 3998, 5092, 6545, 8263, 10486, 13165, 16562, 20630, 25773, 31897, 39546, 48692, 59960, 73423, 89937, 109553, 133439, 161840, 196168, 236843, 285816, 343667, 412950, 494702, 592063, 706671
Offset: 1

Views

Author

Vladeta Jovovic, Feb 16 2004

Keywords

Comments

Row sums of triangle A220504. - Omar E. Pol, Jan 19 2013

Examples

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]. 1 appears 4 times in the first, 1 twice in the second, 2 twice in the third, etc.; thus a(4)=4+2+2+1+1=10.

Links

Crossrefs

Cf. A092314, A092322, A092309, A092321, A092313, A092310, A092311, A092268, A006128, A195053.
For higher-order spt functions see A221140-A221144.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 16 2013
  • Mathematica
    terms = 47; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-François Alcover, Jan 17 2013 *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)
  • PARI
    N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, x^n/(1-x^n) * prod(k=n,N, 1/(1-x^k) )  );
v = Vec(gf)
/* Joerg Arndt, Jan 12 2013 */

Formula

G.f.: Sum_{n>=1} x^n/(1-x^n) * Product_{k>=n} 1/(1-x^k).
a(n) = A000070(n-1) + A195820(n). - Omar E. Pol, Oct 19 2011
a(n) = n*p(n) - N_2(n)/2 = n*A000041(n) - A220908(n)/2 = A066186(n) - A220907(n) = (A220909(n) - A220908(n))/2 = A211982(n)/2 (from Andrews's paper and Garvan's paper). - Omar E. Pol, Jan 03 2013
a(n) = A000041(n) + A000070(n-2) + A220479(n), n >= 2. - Omar E. Pol, Feb 16 2013
Asymptotics (Bringmann-Mahlburg, 2009): a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(8*n)) ~ sqrt(6*n)*A000041(n)/Pi. - Vaclav Kotesovec, Jul 30 2017

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A183011 (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1.

Original entry on oeis.org

-1, 23, 94, 213, 475, 833, 1573, 2505, 4202, 6450, 10038, 14728, 22099, 31411, 45225, 63184, 88473, 120879, 165935, 222950, 300333, 398376, 528054, 691505, 905625, 1172842, 1517628, 1947470, 2494778, 3172675, 4029276, 5083606, 6403683, 8023113
Offset: 0

Views

Author

Omar E. Pol, Jan 21 2011

Keywords

Comments

a(n) is also Tr(n), the numerator of the finite algebraic formula for the number of partitions of n, if n >= 1. The formula is p(n) = Tr(n)/(24*n - 1), n >= 1. See theorem 1.1 of the Bruinier-Ono paper in the link. For the denominators see A183010.
a(n) is also the coefficient of the second term (the trace) in the n-th Bruinier-Ono "partition polynomial" H_n(x), if n >= 1. See the Bruinier-Ono paper, theorem 1.1 and chapter 5 "Examples". For the coefficients of the 4th terms see A187218. - Omar E. Pol, Jul 10 2011
In the Bruinier-Ono-Sutherland paper (Jan 23 2013) partition polynomials are called "partition class polynomials". See also Sutherland's table of Hpart_n(x) in link section. - Omar E. Pol, Feb 20 2013

Examples

			1. For n = 6, the number of partitions of 6 is 11, so a(6) = (24*6 - 1)*11 = 143*11 = 1573.
2. For n = 1, in the Bruinier-Ono paper, chapter 5, the first "partition polynomial" is H_1(x) = x^3 - 23*x^2 + (3592/23)*x - 419. The coefficient of the second term (the trace) is 23, so a(1) = 23.
G.f. = -1 + 23*x + 94*x^2 + 213*x^3 + 475*x^4 + 833*x^5 + 1573*x^6 + 2505*x^7 + ...
G.f. = -q^-1 + 23*q^23 + 94*q^47 + 213*q^71 + 475*q^95 + 833*q^119 + 1573*q^143 + ...

Links

Crossrefs

Positive terms are the partial sums of A183012, also the column 24 of A182729.
Cf. A000041, A008606, A066186, A183006, A183009, A183010, A183054, A187206.

Programs

  • Mathematica
    a[ n_] := (24 n - 1) SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)
  • PARI
    {a(n) = if( n<0, 0, (24*n - 1) * numbpart(n))}; /* Michael Somos, Aug 28 2013 */

Formula

a(n) = A183010(n)*A000041(n).
a(n) = 24*A066186(n) - A000041(n) = A183009(n) - A000041(n) = (A008606(n)-1)*A000041(n).
a(n) = 12M_2(n) - p(n) = 24spt(n) + 12N_2(n) - p(n) = 12*A220909(n) - A000041(n) = 24*A092269(n) + 12*A220908(n) - A000041(n), n >= 1. - Omar E. Pol, Feb 17 2013
G.f.: Sum_{k >= 0} a(k) * q^(24*k - 1) = q * d/dq (1/q * Product_{k > 0} 1 / (1 - q^(24*k))). - Michael Somos, Aug 28 2013

A220909 The second crank moment function M_2(n).

Original entry on oeis.org

0, 2, 8, 18, 40, 70, 132, 210, 352, 540, 840, 1232, 1848, 2626, 3780, 5280, 7392, 10098, 13860, 18620, 25080, 33264, 44088, 57730, 75600, 97900, 126672, 162540, 208208, 264770, 336240, 424204, 534336, 669438, 837080, 1041810, 1294344, 1601138, 1977140, 2432430, 2987040, 3655806
Offset: 0

Views

Author

N. J. A. Sloane, Jan 02 2013

Keywords

Comments

M_2(n) is defined to be Sum_{m=-n..n} m^2 M(m,n) where M(m,n) is the number of partitions of n with crank m except for n=1 where M(-1,1) = M(1,1) = -M(0,1) = 1. - Michael Somos, Nov 10 2013
From Omar E. Pol, Jul 25 2022: (Start)
Apart from the initial zero this is also:
Convolution of A074400 and A000041.
Convolution of A000203 and A139582. (End)

Examples

			G.f. = 2*x + 8*x^2 + 18*x^3 + 40*x^4 + 70*x^5 + 132*x^6 + 210*x^7 + ...
For n=1, M_2(1) = Sum_{m=-1..1} m^2 * M(m,2) = (-1)^2*1 + 0^2*(-1) + 1^2*1 = 2. For n=2, the partition [2] has crank 2 and partition [1,1] has crank -2, hence M_2(2) = 2^2 + (-2)^2 = 8. - _Michael Somos_, Nov 10 2013

Links

Crossrefs

Cf. A000041, A066186, A092269, A220908.
Cf. A000203, A074400, A139582.

Programs

  • Mathematica
    a[ n_] := 2 n PartitionsP @ n (* Michael Somos, Nov 10 2013 *)
  • PARI
    {a(n) = if( n<0, 0, 2 * n * polcoeff( 1 / eta(x + x * O(x^n)), n))} /* Michael Somos, Nov 10 2013 */

Formula

a(n) = 2*n*A000041(n) = 2*A066186(n).
a(n) = n*A139582(n). - Omar E. Pol, Jan 03 2013
a(n) = A220908(n) + A211982(n), n >= 1. - Omar E. Pol, Jan 17 2013
a(n) = 2*(A092269(n) + A220907(n)), n >= 1. _Omar E. Pol, Feb 18 2013
a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016

A183009 a(n) = 24*n*p(n) = 24*n*A000041(n).

Original entry on oeis.org

24, 96, 216, 480, 840, 1584, 2520, 4224, 6480, 10080, 14784, 22176, 31512, 45360, 63360, 88704, 121176, 166320, 223440, 300960, 399168, 529056, 692760, 907200, 1174800, 1520064, 1950480, 2498496, 3177240, 4034880, 5090448, 6412032
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2011

Keywords

Comments

a(n) is also the sum of the partition number of n and the "trace" Tr(n) of A183011. a(n) = p(n) + Tr(n).
a(n) is also the number of "sectors" or "half-periods" in all partitions of n in some versions of the shell model of partitions of A135010.

Examples

			The number of partitions of 6 is p(6) = A000041(6) = 11, so a(6) = 24*6*11 = 1584.
Also the trace Tr(6) = A183011(6) = 1573, so a(6) = p(6) + Tr(6) = 11 + 1573 = 1584.

Crossrefs

Partial sums of A183006. Column 24 of A182728.
Cf. A000041, A000796, A008606, A018253, A066186, A135010, A182742, A182743, A183008, A183010, A183011.

Programs

  • Mathematica
    Table[24n*PartitionsP[n],{n,40}] (* Harvey P. Dale, Mar 07 2019 *)

Formula

a(n) = A008606(n)*A000041(n) = 24*A066186(n) = n*A183008(n).
a(n) = p(n) + Tr(n) = A000041(n) + A183011(n).
a(n) = 12*M_2(n) = 24*spt(n) + 12*N_2(n) = 12*A220909(n) = 24*A092269(n) + 12*A220908(n). - Omar E. Pol, Feb 17 2013

A211982 Second crank moment minus second rank moment: M_2(n) - N_2(n) = 2*spt(n).

Original entry on oeis.org

2, 6, 10, 20, 28, 52, 70, 114, 160, 238, 322, 476, 630, 880, 1178, 1602, 2096, 2814, 3640, 4798, 6174, 7996, 10184, 13090, 16526, 20972, 26330, 33124, 41260, 51546, 63794, 79092, 97384, 119920, 146846, 179874, 219106, 266878, 323680, 392336, 473686
Offset: 1

Views

Author

Omar E. Pol, Jan 03 2013

Keywords

Comments

Also total number of smallest parts in all partitions of n, multiplied by 2.

Links

Crossrefs

Cf. A092269, A066186, A220907, A220908, A220909.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> 2* b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 17 2013
  • Mathematica
    terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; 2*CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-François Alcover, Jan 17 2013, from 2nd formula *)

Formula

a(n) = A220909(n) - A220908(n) = 2*A092269(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(2*n)) * (1 - Pi/(24*sqrt(6*n)) + (144+Pi^2)/(6912*n)). - Vaclav Kotesovec, Jul 31 2017

A220907 a(n) = n*p(n)-spt(n) (= n*A000041(n) - A092269(n)).

Original entry on oeis.org

0, 1, 4, 10, 21, 40, 70, 119, 190, 301, 455, 686, 998, 1450, 2051, 2895, 4001, 5523, 7490, 10141, 13545, 18046, 23773, 31255, 40687, 52850, 68105, 87542, 111755, 142347, 180205, 227622, 286027, 358580, 447482, 557235, 691016, 855131, 1054375, 1297352, 1591060
Offset: 1

Views

Author

N. J. A. Sloane, Jan 02 2013

Keywords

Links

Crossrefs

Cf. A000041, A092269. Equals A220908/2.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i<1, 0,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> n*combinat[numbpart](n)- b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 09 2013
  • Mathematica
    terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; spt = CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest; a[n_] := n*PartitionsP[n] - spt[[n]]; Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Jan 17 2013, after g.f. of spt(n) *)

Formula

a(n) = N_2(n)/2 (where N_2(n) is the second rank moment, see relation (1.1) of the Garvan reference).
a(n) = A066186(n) - A092269(n). - Omar E. Pol, Jan 08 2013
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)) * (1 - (3*sqrt(6)/(2*Pi) + Pi/(24 * sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jul 31 2017

A211609 12 times the total number of smallest parts in all partitions of n, with a(0) = 0.

Original entry on oeis.org

0, 12, 36, 60, 120, 168, 312, 420, 684, 960, 1428, 1932, 2856, 3780, 5280, 7068, 9612, 12576, 16884, 21840, 28788, 37044, 47976, 61104, 78540, 99156, 125832, 157980, 198744, 247560, 309276, 382764, 474552, 584304, 719520, 881076, 1079244, 1314636, 1601268, 1942080, 2354016, 2842116
Offset: 0

Views

Author

Omar E. Pol, Feb 16 2013

Keywords

Comments

The product 12spt(n) appears in the formula b(n) = 12spt(n)+(24n-1)p(n) which is mentioned in several papers (see Ono's paper, see also Garvan's papers and Garvan's slides in link section). Note that b(n) is A220481(n).
Observation: first 13 terms coincide with the differences between all terms mentioned in a table of special mock Jacobi forms and the first 13 terms of A183011. For the table see Dabholkar-Murthy-Zagier paper, appendix A.1, table of Q_M (weight 2 case), M = 6, C_M = 12. See also the table in page 46. Question: do all terms coincide?

Links

Crossrefs

Cf. A000041, A092269, A183010, A183011, A211982, A220481, A220908, A220909.

Formula

a(n) = A220481(n) - A183011(n).
a(n) = 12spt(n) = 12*A092269(n) = 6(M_2(n) - N_2(n)) = 6*A211982(n) = 6*(A220909(n) - A220908(n)), n >= 1.

A220481 a(n) = 12spt(n) + (24n - 1)p(n), with a(0) = -1.

Original entry on oeis.org

-1, 35, 130, 273, 595, 1001, 1885, 2925, 4886, 7410, 11466, 16660, 24955, 35191, 50505, 70252, 98085, 133455, 182819, 244790, 329121, 435420, 576030, 752609, 984165, 1271998, 1643460, 2105450, 2693522, 3420235, 4338552, 5466370, 6878235, 8607417
Offset: 0

Views

Author

Omar E. Pol, Jan 14 2013

Keywords

Comments

Observation: first 13 terms coincide with all terms mentioned in a table of special mock Jacobi forms. See the Dabholkar-Murthy-Zagier paper, appendix A.1, table of Q_M (weight 2 case), M = 6, C_M = 12. See also the table in page 46.
Question: do all terms coincide?
The formula 12spt(n) + (24n - 1)p(n) is mentioned in several papers (see Ono's paper, see also Garvan's papers and Garvan's slides in link section).
Also a(n) = 12spt + Tr(n), where Tr(n) is the numerator of the Bruinier-Ono formula for the number of partitions of n, if n >= 1 (see theorem 1.1 in the Bruinier-Ono paper). Tr(n) is also the trace of the partition class polynomial Hpart_n(x). For more information see A183011.

Links

Crossrefs

Cf. A000041, A092269, A183010, A183011, A211609, A220908.

Formula

a(n) = 12spt(n) + Tr(n) = 12(3spt(n) + N_2(n)) - p(n), n >= 1.
a(n) = A211609(n) + A183011(n) = 12*A092269(n) + A183011(n) = 12*A092269(n) + A183010(n)*A000041(n) = 12(3*A092269(n) + A220908(n)) - A000041(n), n >= 1.

A211977 Triangle read by rows, T(n,k) = spt_k(n), in which column k lists the k-th order spt-function (without the first k-1 zeros), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 10, 15, 7, 1, 14, 35, 28, 9, 1, 26, 75, 85, 45, 11, 1, 35, 140, 217, 166, 66, 13, 1, 57, 259, 497, 505, 287, 91
Offset: 1

Views

Author

Omar E. Pol, Jan 03 2013

Keywords

Comments

The triangle arises from a table due to F. G. Garvan (See the Garvan's paper, page 19).

Examples

			Triangle begins:
   1;
   3,   1;
   5,   5,   1;
  10,  15,   7,   1;
  14,  35,  28,   9,   1;
  26,  75,  85,  45,  11,   1;
  35, 140, 217, 166,  66,  13,  1;
  ...

Links

Crossrefs

Columns 1-6: A092269, A221140, A221141, A221142, A221143, A221144.
Cf. A000041, A220908, A220909.
Showing 1-10 of 10 results.

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