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

A045931 Number of partitions of n with equal number of even and odd parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 5, 7, 9, 11, 16, 18, 25, 28, 41, 44, 62, 70, 94, 107, 140, 163, 207, 245, 302, 361, 440, 527, 632, 763, 904, 1090, 1285, 1544, 1812, 2173, 2539, 3031, 3538, 4202, 4896, 5793, 6736, 7934, 9221, 10811, 12549, 14661, 16994, 19780
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - Emeric Deutsch, Mar 30 2006

Examples

			a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].
From _Gus Wiseman_, Jan 23 2022: (Start)
The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):
  ()  .  .  21   .  32   2211   43   3221   54       3322   65       4332
                    41          52   4211   63       4321   74       4431
                                61          72       4411   83       5322
                                            81       5221   92       5421
                                            222111   6211   A1       6321
                                                            322211   6411
                                                            422111   7221
                                                                     8211
                                                                     22221111
(End)

Links

Crossrefs

The version for subsets of {1..n} is A001405.
Dominated by A027187 (partitions of even length).
More odd/even parts: A108950/A108949.
More or same number of odd/even parts: A130780/A171966.
The strict case is A239241.
This is column k = 0 of the triangle A240009.
Counting only distinct parts gives A241638, ranked by A325700.
A half-conjugate version is A277579.
These partitions are ranked by A325698.
A000041 counts integer partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A257991/A257992 count odd/even parts by Heinz number.
Cf. A000070, A000700, A000712, A027193, A035363, A097613, A195017.

Programs

  • Maple
    g:=1/product((1-t*x^(2*j-1))*(1-s*x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,56)): P[0]:=1: for n from 1 to 53 do P[n]:=subs(s=1/t,coeff(gser,x^n)) od: seq(coeff(t*P[n],t),n=0..53); # Emeric Deutsch, Mar 30 2006
  • Mathematica
    p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
(* Peter J. C. Moses, Mar 10 2014 *)
nmax = 100; CoefficientList[Series[Sum[x^(3*k) / Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2025 *)

Formula

G.f.: Sum_{k>=0} x^(3*k)/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 18 2007
a(n) = A000041(n)-A171967(n) = A130780(n)-A108950(n) = A171966(n)-A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = A000041(n) - A108950(n) - A108949(n) = A130780(n) + A171966(n) - A000041(n). - Gus Wiseman, Jan 23 2022
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (48*n^(3/2)). - Vaclav Kotesovec, Jun 15 2025

A171966 Number of partitions of n having no more odd than even parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 4, 6, 8, 12, 15, 21, 28, 37, 49, 63, 83, 105, 138, 171, 223, 275, 353, 433, 551, 673, 846, 1031, 1282, 1558, 1922, 2327, 2848, 3440, 4179, 5032, 6078, 7293, 8763, 10482, 12534, 14943, 17797, 21146, 25090, 29719, 35138, 41493, 48908, 57578
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 21 2010

Keywords

Comments

a(n) = A108949(n) + A045931(n) = A000041(n) - A108950(n).
a(n) = Sum_{k=-floor(n/2)+(n mod 2)..0} A240009(n,k). - Alois P. Heinz, Mar 30 2014

Links

Crossrefs

Cf. A130780, A171967, A240009.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t<=0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014
  • Mathematica
    $RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t <= 0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

A130780 Number of partitions of n such that number of odd parts is greater than or equal to number of even parts.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 8, 12, 16, 23, 32, 42, 58, 75, 102, 131, 173, 220, 288, 363, 466, 587, 743, 929, 1164, 1448, 1797, 2224, 2738, 3368, 4122, 5042, 6133, 7466, 9035, 10941, 13184, 15888, 19064, 22876, 27343
Offset: 0

Views

Author

Vladeta Jovovic, Aug 19 2007

Keywords

Comments

a(n) = A108950(n) + A045931(n) = A000041(n) - A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = Sum_{k=0..n} A240009(n,k). - Alois P. Heinz, Mar 30 2014

Examples

			a(5)=6 because we have 5,41,32,311,211 and 11111 (221 does not qualify).

Links

Crossrefs

Cf. A045931, A108949, A108950.
Cf. A171966, A171967. - Reinhard Zumkeller, Jan 21 2010

Programs

  • Maple
    g:=sum(x^k/(product((1-x^(2*i))^2,i=1..k)),k=0..50): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..40); # Emeric Deutsch, Aug 24 2007
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t>=0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014
  • Mathematica
    $RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t >= 0, 1, 0],  If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t + (2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 12 2015, after Alois P. Heinz *)
opgQ[n_]:=Module[{len=Length[n],op},op=Length[Select[n,OddQ]];op>= len-op]; Table[Count[IntegerPartitions[n],?(opgQ)],{n,0,50}] (* _Harvey P. Dale, Dec 12 2021 *)

Formula

G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(2*i))^2.

Extensions

More terms from Emeric Deutsch, Aug 24 2007

A240009 Number T(n,k) of partitions of n, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 3, 2, 2, 2, 1, 1, 0, 1, 1, 1, 2, 2, 2, 4, 3, 2, 2, 1, 1, 0, 1, 1, 2, 4, 5, 3, 4, 4, 2, 2, 1, 1, 0, 1, 1, 1, 2, 3, 3, 5, 7, 5, 4, 4, 2, 2, 1, 1, 0, 1, 1, 2, 4, 7, 7, 6, 8, 6, 4, 4, 2, 2, 1, 1, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Mar 30 2014

Keywords

Comments

T(n,k) = T(n+k,-k).
Sum_{k=-floor(n/2)+(n mod 2)..-1} T(n,k) = A108949(n).
Sum_{k=-floor(n/2)+(n mod 2)..0} T(n,k) = A171966(n).
Sum_{k=1..n} T(n,k) = A108950(n).
Sum_{k=0..n} T(n,k) = A130780(n).
Sum_{k=-1..1} T(n,k) = A239835(n).
Sum_{k<>0} T(n,k) = A171967(n).
T(n,-1) + T(n,1) = A239833(n).
Sum_{k=-floor(n/2)+(n mod 2)..n} k * T(n,k) = A209423(n).
Sum_{k=-floor(n/2)+(n mod 2)..n} (-1)^k*T(n,k) = A081362(n) = (-1)^n*A000700(n).

Examples

			T(5,-1) = 1: [2,2,1].
T(5,0) = 2: [4,1], [3,2].
T(5,1) = 1: [5].
T(5,2) = 1: [2,1,1,1].
T(5,3) = 1: [3,1,1].
T(5,5) = 1: [1,1,1,1,1].
Triangle T(n,k) begins:
: n\k : -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 ...
+-----+----------------------------------------------------
:  0  :                 1;
:  1  :                    1;
:  2  :              1, 0, 0, 1;
:  3  :                 1, 1, 0, 1;
:  4  :           1, 1, 0, 1, 1, 0, 1;
:  5  :              1, 2, 1, 1, 1, 0, 1;
:  6  :        1, 1, 1, 1, 2, 2, 1, 1, 0, 1;
:  7  :           1, 2, 3, 2, 2, 2, 1, 1, 0, 1;
:  8  :     1, 1, 2, 2, 2, 4, 3, 2, 2, 1, 1, 0, 1;
:  9  :        1, 2, 4, 5, 3, 4, 4, 2, 2, 1, 1, 0, 1;
: 10  :  1, 1, 2, 3, 3, 5, 7, 5, 4, 4, 2, 2, 1, 1, 0, 1;

Links

Crossrefs

Columns k=(-1)-10 give: A239832, A045931, A240010, A240011, A240012, A240013, A240014, A240015, A240016, A240017, A240018, A240019.
Row sums give A000041.
T(2n,n) gives A002865.
T(4n,2n) gives A182746.
T(4n+2,2n+1) gives A182747.
Row lengths give A016777(floor(n/2)).
Cf. A240021 (the same for partitions into distinct parts), A242618 (the same for parts counted without multiplicity).
Cf. A000700, A081362, A209423.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i)*x^(2*irem(i, 2)-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):
seq(T(n), n=0..14);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]*x^(2*Mod[i, 2]-1)]]]; T[n_] := (degree = Exponent[b[n, n], x]; ldegree = -Exponent[b[n, n] /. x -> 1/x, x]; Table[Coefficient[b[n, n], x, i], {i, ldegree, degree}]); Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *)
  • PARI
    N=20; q='q+O('q^N);
e(n) = if(n%2!=0, u, 1/u);
gf = 1 / prod(n=1,N, 1 - e(n)*q^n );
V = Vec( gf );
{ for (j=1, #V,  \\ print triangle, including leading zeros
    for (i=0, N-j, print1("   "));  \\ padding
    for (i=-j+1, j-1, print1(polcoeff(V[j], i, u),", "));
    print();
); }
/* Joerg Arndt, Mar 31 2014 */

Formula

G.f.: 1 / prod(n>=1, 1 - e(n)*q^n ) = 1 + sum(n>=1, e(n)*q^n / prod(k=1..n, 1-e(k)*q^k) ) where e(n) = u if n odd, otherwise 1/u; see Pari program. [Joerg Arndt, Mar 31 2014]

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 4, 6, 15, 16, 21, 24, 25, 35, 60, 64, 77, 84, 90, 91, 96, 100, 121, 126, 140, 143, 150, 210, 221, 240, 247, 256, 289, 297, 308, 323, 336, 351, 360, 364, 375, 384, 400, 437, 462, 484, 490, 495, 504, 525, 529, 546, 551, 560, 572, 585, 600, 625, 667, 686, 726
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.
These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

Examples

			The terms and their prime indices begin:
    1: ()
    4: (1,1)
    6: (2,1)
   15: (3,2)
   16: (1,1,1,1)
   21: (4,2)
   24: (2,1,1,1)
   25: (3,3)
   35: (4,3)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
   77: (5,4)
   84: (4,2,1,1)
   90: (3,2,2,1)
   91: (6,4)
   96: (2,1,1,1,1,1)

Crossrefs

A subset of A028260 (even bigomega), counted by A027187.
These partitions are counted by A277579.
This is the half-conjugate version of A325698, counted by A045931.
A000041 counts partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A100824 counts partitions with at most one odd part, ranked by A349150.
A108950/A108949 count partitions with more odd/even parts.
A122111 represents conjugation using Heinz numbers.
A130780/A171966 count partitions with more or equal odd/even parts.
A257991/A257992 count odd/even prime indices.
A316524 gives the alternating sum of prime indices (reverse: A344616).
Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

Formula

A257992(a(n)) = A257991(A122111(a(n))).

A098123 Number of compositions of n with equal number of even and odd parts.

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 6, 6, 24, 28, 60, 130, 190, 432, 770, 1386, 2856, 5056, 9828, 18918, 34908, 68132, 128502, 244090, 470646, 890628, 1709136, 3271866, 6238986, 11986288, 22925630, 43932906, 84349336, 161625288, 310404768, 596009494
Offset: 0

Views

Author

Vladeta Jovovic, Sep 24 2004

Keywords

Examples

			From _Gus Wiseman_, Jun 26 2022: (Start)
The a(0) = 1 through a(7) = 6 compositions (empty columns indicated by dots):
  ()  .  .  (12)  .  (14)  (1122)  (16)
            (21)     (23)  (1212)  (25)
                     (32)  (1221)  (34)
                     (41)  (2112)  (43)
                           (2121)  (52)
                           (2211)  (61)
(End)

Links

Crossrefs

For partitions: A045931, ranked by A325698, strict A239241 (conj A352129).
Column k=0 of A242498.
Without multiplicity: A242821, for partitions A241638 (ranked by A325700).
These compositions are ranked by A355321.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000700, A000712, A001405, A026424, A027193, A028260, A097613, A240009, A277579 (ranked by A349157).
Cf. A025178.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Count[#,?EvenQ]==Count[#,?OddQ]&]],{n,0,15}] (* Gus Wiseman, Jun 26 2022 *)

Formula

a(n) = Sum_{k=floor(n/3)..floor(n/2)} C(2*n-4*k,n-2*k)*C(n-1-k,2*n-4*k-1).
Recurrence: n*(2*n-7)*a(n) = 2*(n-2)*(2*n-5)*a(n-2) + 2*(2*n-7)*(2*n-3)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - Vaclav Kotesovec, May 01 2014
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.94696532812840456026081823863... is the root of the equation 1-4*d-2*d^2+d^4 = 0, c = 0.225563290820392765554898545739... is the root of the equation 43*c^4-18*c^2+8*c-1=0. - Vaclav Kotesovec, May 01 2014
G.f.: 1/sqrt(1 - 4*x^3/(1-x^2)^2). - Seiichi Manyama, May 01 2025

A108950 Number of partitions of n with more odd parts than even parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 9, 14, 18, 27, 35, 49, 64, 86, 113, 148, 192, 247, 319, 404, 517, 649, 822, 1024, 1285, 1590, 1979, 2436, 3007, 3682, 4515, 5501, 6703, 8131, 9851, 11899, 14344, 17252, 20703, 24804, 29640, 35377, 42115, 50085, 59415, 70420, 83261, 98365, 115947, 136557
Offset: 1

Views

Author

Len Smiley, Jul 21 2005

Keywords

Examples

			a(4) = 3: {[3,1], [2,1,1], [1,1,1,1]}; a(5) = 4: {[5], [3,1,1], [2,1,1,1], [1,1,1,1,1]}.

Links

Crossrefs

Cf. A045931 for #even parts = #odd parts, A108949 for #even parts > #odd parts.
Cf. A171966, A171967. - Reinhard Zumkeller, Jan 21 2010

Programs

  • Maple
    with(combinat,partition):oddbigrevn:=proc(n::nonnegint) local evencount,oddcount,bigcount,parts,i,j; printlevel:=-1;bigcount:=0; partitions:=partition(n);for i from 1 to nops(partitions) do evencount:=0; oddcount:=0;for j from 1 to nops(partitions[i]) do if (op(j,partitions[i]) mod 2 <>0) then oddcount:=oddcount+1 fi; if (op(j,partitions[i]) mod 2 =0) then evencount:=evencount+1 fi od; if (evencount0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 30 2014
  • Mathematica
    p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] > Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 15}] (* partitions of n with # odd parts > # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 1, 30}] (* A108950 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t>0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t + (2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=0} x^k*(1-x^(2*k))/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 19 2007
a(n) = A130780(n) - A045931(n) = A171967(n) - A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = Sum_{k=1..n} A240009(n,k). - Alois P. Heinz, Mar 30 2014
G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2-n)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2. - Jeremy Lovejoy, Jan 12 2021

Extensions

More terms from Joerg Arndt, Oct 04 2012

A171967 Number of partitions of n with distinct numbers of odd and even parts.

Original entry on oeis.org

0, 1, 2, 2, 5, 5, 10, 12, 20, 25, 37, 49, 68, 90, 119, 158, 206, 269, 344, 446, 565, 722, 908, 1148, 1435, 1795, 2229, 2765, 3416, 4204, 5164, 6315, 7717, 9380, 11406, 13793, 16692, 20093, 24203, 29012, 34799, 41552, 49636, 59059, 70279, 83341, 98822
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 21 2010

Keywords

Comments

a(n) = A000041(n) - A045931(n) = A108949(n) + A108950(n).
a(n) = Sum_{k<>0} A240009(n,k). - Alois P. Heinz, Mar 30 2014

Links

Crossrefs

Cf. A130780, A171966.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t<>0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014
  • Mathematica
    $RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t != 0, 1, 0], If[i < 1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

A322014 Heinz numbers of integer partitions with an even number of even parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 20, 21, 22, 23, 25, 31, 32, 34, 36, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 55, 57, 59, 62, 64, 67, 68, 72, 73, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 91, 92, 94, 97, 98, 99, 100, 103, 105, 109, 110, 111, 114, 115, 118
Offset: 1

Views

Author

Gus Wiseman, Nov 24 2018

Keywords

Comments

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

Links

Crossrefs

Cf. A000701, A046682, A056239, A058695, A058696, A108949, A296150, A321753.

Programs

  • Maple
    a:= proc(n) option remember; local k; for k from 1+`if`(n=1,
      0, a(n-1)) while add(`if`(numtheory[pi](i[1])::odd,
      0, i[2]), i=ifactors(k)[2])::odd do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 24 2018
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],EvenQ[Count[primeMS[#],_?EvenQ]]&]
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