A015128 -id:A015128 - cp's OEIS frontend

A356290 a(n) = Sum_{k=0..n} binomial(3n, n-k) v(k), where v(k) is the number of overpartitions of n (A015128).

1, 5, 31, 200, 1309, 8627, 57082, 378648, 2516111, 16740913, 111494801, 743137984, 4956359312, 33074272702, 220810039566, 1474764797488, 9853307017341, 65853733243281, 440255398634199, 2944041287677060, 19691951641479427, 131744163990056479, 881586559906575688

Offset: 0

Vaclav Kotesovec, Aug 02 2022

nonn

Cf. A015128, A266497, A356280, A356281, A356282, A356283, A356289.

Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]

a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} v(j)/2^j = 8.2559879357782500655441408494322731265270016167882303456037...

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

N. J. A. Sloane

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

			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)

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.

David W. Wilson, Table of n, a(n) for n = 0..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 836. [scanned copy]
Scott Ahlgren and Ken Ono, Addition and Counting: The Arithmetic of Partitions, Notices of the AMS, 48 (2001) pp. 978-984.
Scott Ahlgren and Ken Ono, Congruence properties for the partition function
Scott Ahlgren and Ken Ono, Congruence properties for the partition function, PNAS, vol. 98 no. 23, 12882-12884.
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
Gert Almkvist and Herbert S. Wilf, On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n). [Broken link?]
Gert Almkvist and Herbert S. Wilf, On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n), Journal of Number Theory, Vol. 50, No. 2, 1995, pp. 329-334.
Amazing Mathematical Object Factory, Information on Partitions. [Wayback Machine link]
Edward Anderson, Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces, arXiv:1805.03346 [gr-qc], 2018.
Theresa C. Anderson, Larry Rolen and Ruth Stoehr, Benford's Law for Coefficients of Modular Forms and Partition Functions, Proceedings of the American Mathematical Society, Vol. 139, No. 5, May 2011, pp. 1533-1541.
George E. Andrews, Three Aspects of Partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R1.
George E. Andrews and Ken Ono, Ramanujan's congruences and Dyson's crank
George E. Andrews and Ranjan Roy, Ramanujan's Method in q-series Congruences, The Electronic Journal of Combinatorics, Volume 4, Issue 2 (1997) (The Wilf Festschrift volume) > Research Paper #R2.
George E. Andrews, Sumit Kumar Jha, and J. López-Bonilla, Sums of Squares, Triangular Numbers, and Divisor Sums, Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.5.
Anonymous, Bibliography on Partitions
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, An ultimately periodic chain in the integral Lie ring of partitions, J. Algebr. Comb. (2024). See p. 11.
Joerg Arndt, Matters Computational (The Fxtbook), section 16.4, pp.344-353.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
Alexander Berkovich and Frank G. Garvan, On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5, arXiv:math/0401012 [math.CO], 2004.
Alexander Berkovich and Frank G. Garvan, On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations, arXiv:math/0402439 [math.CO], 2004.
Bruce C. Berndt, Ramanujan's congruences for the partition function modulo 5,7 and 11
Bruce C. Berndt and K. Ono, Ramanujan's Unpublished Manuscript On The Partition And Tau Functions With Proofs And Commentary
Bruce C. Berndt and K. Ono, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Frits Beukers, Ramanujan and The Partition Function (text in Dutch), Pythagoras, Wiskundetijdschrift voor Jongeren, 38ste Jaargang, Nummer 6, Agustus 1999, pp. 15-16.
Henry Bottomley, Illustration of initial terms
Henry Bottomley, Illustration of initial terms of A000009, A000041 and A047967
Henry Bottomley, Partition and composition calculator [broken link]
Kevin S. Brown, Additive Partitions of Numbers [Broken link]
Kevin S. Brown, Additive Partitions of Numbers [Cached copy of lost web page]
Kevin S. Brown's Mathpages, Computing the Partitions of n
Jan Hendrik Bruinier, Amanda Folsom, Zachary A. Kent and Ken Ono, Recent work on the partition function
Jan Hendrik Bruinier and Ken Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
Chao-Ping Chen and Hui-Jie Zhang, Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality, Journal of Inequalities and Applications, 2017.
Yuriy Choliy and Andrew V. Sills, A formula for the partition function that 'counts'
Lynn Chua and Krishanu Roy Sankar, Equipopularity Classes of 132-Avoiding Permutations, The Electronic Journal of Combinatorics 21(1)(2014), #P59. [Cited by Shalosh B. Ekhad and Doron Zeilberger, 2014] - _N. J. A. Sloane_, Mar 31 2014
CombOS - Combinatorial Object Server, Generate integer partitions
Jimena Davis and Elizabeth Perez, Computations Of The Partition Function, p(n)
Stephen DeSalvo and Igor Pak, Log-Concavity of the Partition Function, arXiv:1310.7982 [math.CO], 2013-2014.
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
Shalosh B. Ekhad and Doron Zeilberger, Automatic Proofs of Asymptotic Abnormality (and much more!) of Natural Statistics Defined on Catalan-Counted Combinatorial Families, arXiv:1403.5664 [math.CO], 2014.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
FindStat - Combinatorial Statistic Finder, Integer partitions
Nathan J. Fine, Some New Results On Partitions
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 41.
Amanda Folsom, Zachary A. Kent and Ken Ono, l-adic properties of the partition function, in press.
Amanda Folsom, Zachary A. Kent and Ken Ono, l-adic properties of the partition function, Adv. Math. 229 (2012) 1586.
B. Forslund, Partitioning Integers
Harald Fripertinger, Partitions of an Integer
Bert Fristedt, The structure of random partitions of large integers, Transactions of the American Mathematical Society, 337.2 (1993): 703-735. [A classic paper - _N. J. A. Sloane_, Aug 27 2018]
GEO magazine, Zahlenspalterei
James Grime and Brady Haran, Partitions, Numberphile video (2016).
Harald Grosse, Alexander Hock, and Raimar Wulkenhaar, A Laplacian to compute intersection numbers on M_(g,n) and correlation functions in NCQFT, arXiv:1903.12526 [math-ph], 2019.
G. H. Hardy and S. Ramanujan, Asymptotic formulas in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-115.
A. Hassen and T. J. Olsen, Playing With Partitions On The Computer
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
Alexander D. Healy, Partition Identities
Ferdinand Ihringer, Remarks on the Erdős Matching Conjecture for Vector Spaces, arXiv:2002.06601 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 61 and Encyclopedia of Combinatorial Structures 74
Fredrik Johansson, Fast arbitrary-precision evaluation of special functions in the Arb library, OPSFA13, NIST, June 2015, page 15.
Jonthan M. Kane, Distribution of orders of Abelian groups, Math. Mag., 49 (1976), 132-135.
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014.
Erica Klarreich, Pieces of Numbers: A proof brings closure to a dramatic tale of partitions and primes, Science News, Week of Jun 18, 2005; Vol. 167, No. 25, p. 392.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
J. Laurendi, Partitions of Integers
Oleg Lazarev, Matt Mizuhara, and Ben Reid, Some Results in Partitions, Plane Partitions, and Multipartitions
Li Wenwei, Estimation of the Partition Number: After Hardy and Ramanujan, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
T. Lockette, Explore Magazine, Path To Partitions
Jerome Malenfant, Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers, arXiv:1103.1585 [math.NT], 2011.
Dr. Math, Partitioning the Integers and Partitioning an Integer
M. MacMahon, Collected Papers of Ramanujan, Table for p(n);n=1 through 200
S. Markovski and M. Mihova, An explicit formula for computing the partition numbers p(n), Math. Balkanica 22 (2008) 101-119 MR2467361
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.
Johannes W. Meijer and Manuel Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187.
Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.
Mircea Merca and M. D. Schmidt, The partition function p(n) in terms of the classical Mobius function, Ramanujan J. 49 (1) (2019) 87-96.
István Mező, Several special values of Jacobi theta functions arXiv:1106.2703v3 [math.CA], 2011-2013.
Gerard P. Michon, Table of partition function p(n) (n=0 through 4096)
Gerard P. Michon, Partition function
G. A. Miller, Number of the abelian groups of a given order
Hisanori Mishima, Factorization of Partition Numbers
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
D. J. Newman, A simplified proof of the partition formula, Michigan Math. J. 9:3 (1962), pp. 193-287.
Jean-Louis Nicolas, Sur les entiers N pour lesquels il y a beaucoup de groupes abéliens d'ordre N, Annales de l'Institut Fourier, Tome 28 (1978) no. 4, p. 1-16.
OEIS Wiki, Sorting numbers
Ken Ono, Arithmetic of the partition function
Ken Ono, Parity of the partition function, Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 35-42.
Ken Ono, Distribution of the partition function modulo m, Annals Math. 151 (2000), 293-307.
Ken Ono (with J. Bruinier, A. Folsom and Z. Kent), Emory University, Adding and counting
T. J. Osler, Playing with Partitions on the Computer
I. Pak, Partition bijections, a survey, Ramanujan J. 12 (2006) 5-75.
Michael Penn, Rogers-Ramanujan Identities, Youtube playlist, 2019, 2020.
I. Peterson, The Power Of Partitions
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Michel Planat, Quantum 1/f Noise in Equilibrium: from Planck to Ramanujan, arXiv:math-ph/0307033, 2003.
M. Presern, Some Results On Partitions
W. A. Pribitkin, Revisiting Rademacher's Formula for the Partition Function p(n), The Ramanujan Journal 4(4) 2000.
Srinivasa Ramanujan, Some Properties Of p(n), The Number Of Partitions Of n
Srinivasa Ramanujan, Congruence Properties Of Partitions
Srinivasa Ramanujan, Congruence Properties Of Partitions
Srinivasa Ramanujan and G. H. Hardy, Une formule asymptotique pour le nombre de partitions de n
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. D. Rosenhouse, Partitions of Integers
J. D. Rosenhouse, Solutions to Problems
Kate Rudolph, Pattern Popularity in 132-Avoiding Permutations, The Electronic Journal of Combinatorics 20(1)(2013), #P8. [Cited by Shalosh B. Ekhad and Doron Zeilberger, 2014] - _N. J. A. Sloane_, Mar 31 2014
F. Ruskey, Generate Numerical Partitions
F. Ruskey, The first 284547 partition numbers (52MB compressed file, archived link)
M. Savic, The Partition Function and Ramanujan's 5k+4 Congruence
Zhumagali Shomanov, Combinatorial formula for the partition function, arXiv:1508.03173 [math.CO], 2015.
T. Sillke, Number of integer partitions
R. P. Stanley, A combinatorial miscellany
Cormac O'Sullivan, Detailed asymptotic expansions for partitions into powers, arXiv:2205.13468 [math.NT], 2022-3.
Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595 [math.CO], 2013.
R. L. Weaver, New Congruences for the Partition Function, The Ramanujan Journal 5(1) 2001.
Eric Weisstein's World of Mathematics, Partition, Partition Function P, q-Pochhammer Symbol, and Ramanujan's Identity
West Sussex Grid for Learning, Multicultural Mathematics, Ramanujan's Partition of Numbers
Thomas Wieder, Comment on A000041
Wikipedia, Partition (number theory)
H. S. Wilf, Lectures on Integer Partitions
Wolfram Research, Generating functions of p(n)
D. J. Wright, Partitions [broken link]
Doron Zeilberger, Noam Zeilberger, Two Questions about the Fractional Counting of Partitions, arXiv:1810.12701 [math.CO], 2018.
Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).
Index entries for "core" sequences
Index entries for related partition-counting sequences
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
Index entries for sequences related to rooted trees
Index entries for sequences related to Benford's law

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).

List([1..10],n->Size(OrbitsDomain(SymmetricGroup(IsPermGroup,n),SymmetricGroup(IsPermGroup,n),\^))); # Attila Egri-Nagy, Aug 15 2014

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

# DedekindEta is defined in A000594
A000041List(len) = DedekindEta(len, -1)
A000041List(50) |> println # Peter Luschny, Mar 09 2018

a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];

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

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 *)

num_partitions(60,list); /* Emanuele Munarini, Feb 24 2014 */

combinat::partitions::count(i) $i=0..54 // Zerinvary Lajos, Apr 16 2007

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x * O(x^n)), n))};

/* 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)};
```

{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))};

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

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

use ntheory ":all"; my @p = map { partitions($) } 0..100; say "[@p]"; # _Dana Jacobsen, Sep 06 2015

from sympy.functions.combinatorial.numbers import partition
print([partition(i) for i in range(101)]) # Joan Ludevid, May 25 2025

#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

[number_of_partitions(n) for n in range(46)]  # Zerinvary Lajos, May 24 2009

@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

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(1, 0)
b = EulerTransform(a)
print([b(n) for n in range(50)]) # Peter Luschny, Nov 11 2020

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

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

A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581

Offset: 0

N. J. A. Sloane, Warren D. Smith

nonn

Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - Noureddine Chair, Feb 03 2005
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4. - James Sellers, Feb 08 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
Equals polcoeff inverse of A010054 with alternate signs. - Gary W. Adamson, Mar 15 2010
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012
Convolution of A035363 and A000700. - Vaclav Kotesovec, Aug 17 2015
Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - Seiichi Manyama, Jul 07 2018

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
From _Seiichi Manyama_, Jul 07 2018: (Start)
n | the ways to stack n triangles in a valley
--+------------------------------------------------------
1 | *---*
  |  \ /
  |   *
  |
2 |   *
  |  / \
  | *---*
  |  \ /
  |   *
  |
3 |   *---*     *---*
  |  / \ /       \ / \
  | *---*         *---*
  |  \ /           \ /
  |   *             *
  |
4 |     *                       *
  |    / \                     / \
  |   *---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*
  |  \ /           \ /           \ /
  |   *             *             *
  |
5 |     *---*         *         *         *---*
  |    / \ /         / \       / \         \ / \
  |   *---*     *---*---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |
6 |       *
  |      / \
  |     *---*         *---*     *   *     *---*
  |    / \ /         / \ /     / \ / \     \ / \
  |   *---*     *---*---*     *---*---*     *---*---*
  |  / \ /       \ / \ /       \ / \ /       \ / \ /
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |   *
  |  / \
  | *---*
  |  \ / \
  |   *---*
  |    \ / \
  |     *---*
  |      \ /
  |       *
(End)

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Luca Ferrari, Schröder partitions, Schröder tableaux and weak poset patterns, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
L. Wang, New Congruences for Partitions where the Odd Parts are Distinct, J. Int. Seq. 18 (2015) # 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

See also Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A015128, A046682, A106459.
Cf. A163203.
Cf. A010054, A085642, A316384.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 06 2013

CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *)
CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
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-Mod[i, 2]]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz *)

{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 22 2009
(GW-BASIC)
' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):
10 Dim A000217(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 51: For j = 1 to n
30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j))
40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012

a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - Vladeta Jovovic, Sep 22 2002 [corrected by Vaclav Kotesovec, Aug 17 2015]
a(n) = A059777(n-1)+A059777(n), n > 0. - Vladeta Jovovic, Sep 22 2002
G.f.: Product_{m>=1} (1+x^m)^(if A001511(m) > 1, A001511(m)-1 else A001511(m)). - Jon Perry, Apr 15 2005
Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
Convolution inverse of A106459. - Michael Somos, Nov 02 2005
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - Paul D. Hanna, Jul 22 2009
a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017
Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - Eric M. Schmidt, Sep 21 2017
a(n) = A000041(n) - A085642(n), for n >= 1. - Wouter Meeussen, Dec 20 2017

G.f. and more terms from Vladeta Jovovic, Feb 05 2002

A067661 Number of partitions of n into distinct parts such that number of parts is even.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404

Offset: 0

Naohiro Nomoto, Feb 23 2002

Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700).

			G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(3) = 1 through a(14) = 11 partitions (A-D = 10..13):
  21   31   32   42   43   53   54   64     65     75     76     86
            41   51   52   62   63   73     74     84     85     95
                      61   71   72   82     83     93     94     A4
                                81   91     92     A2     A3     B3
                                     4321   A1     B1     B2     C2
                                            5321   5421   C1     D1
                                                   6321   5431   5432
                                                          6421   6431
                                                          7321   6521
                                                                 7421
                                                                 8321
(End)

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (2).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Dominates A000009.
Numbers with these strict partitions as binary indices are A001969.
The non-strict case is A027187, ranked by A028260.
The Heinz numbers of these partitions are A030229.
The odd version is A067659, ranked by A030059.
The version for rank is A117192, with positive case A101708.
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A008289 counts strict partitions by sum and length.
A026805 counts partitions whose least part is even.
Cf. A000700, A027193, A058696, A072233, A096373, A236913, A244990, A300061.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] + QPochhammer[ x]) / 2, {x, 0, n}]; (* Michael Somos, May 06 2015 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 08 2021 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A) + eta(x + A)) / 2, n))}; /* Michael Somos, Feb 14 2006 */

N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);
gf=sum(n=0, S, (n%2==0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
Vec(gf)  \\ Joerg Arndt, Apr 01 2014

G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^3 + q^4 + 2 q^5 + 2 q^6 + 3 q^7 + ... = Sum_{n >= 0} q^(n(2n+1))/(q; q){2n} [_Bill Gosper, Jun 25 2005]
Also, let B(q) = Sum_{n >= 0} A067659(n) q^n = q + q^2 + q^3 + q^4 + q^5 + 2 q^6 + ... Then B(q) = Sum_{n >= 0} q^((n+1)(2n+1))/(q; q)_{2n+1}.
Also we have the following identity involving 2 X 2 matrices:
Prod_{k >= 1} [ 1, q^k; q^k, 1 ] = [ A(q), B(q); B(q), A(q) ] [Bill Gosper, Jun 25 2005]
a(n) = (A000009(n)+A010815(n))/2. - Vladeta Jovovic, Feb 24 2002
Expansion of (1 + phi(-x)) / (2*chi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006
a(n) + A067659(n) = A000009(n). - R. J. Mathar, Jun 18 2016
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 24 2018
A000009(n) = a(n) + A067659(n). - Gus Wiseman, Jan 09 2021
From Peter Bala, Feb 05 2021: (Start)
G.f.: A(x) = (1/2)*((Product_{n >= 0} 1 + x^n) + (Product_{n >= 0} 1 - x^n)).
Let B(x) denote the g.f. of A067659. Then
A(x)^2 - B(x)^2 = A(x^2) - B(x^2) = Product_{n >= 1} 1 - x^(2*n) = Sum_{n in Z} (-1)^n*x^(n*(3*n+1)).
A(x) + B(x) is the g.f. of A000009.
1/(A(x) - B(x)) is the g.f. of A000041.
(A(x) + B(x))/(A(x) - B(x)) is the g.f. of A015128.
A(x)/(A(x) + B(x)) = Sum_{n >= 0} (-1)^n*x^n^2 = (1 + theta_3(-x))/2.
B(x)/(A(x) - B(x)) is the g.f. of A014968.
A(x)/(A(x^2) - B(x^2)) is the g.f. of A027187.
B(x)/(A(x^2) - B(x^2)) is the g.f. of A027193. (End)

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

Original entry on oeis.org

1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604

Offset: 0

R. J. Mathar, Feb 11 2009

Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Ali H. Al-Saedi, Congruences for restricted plane overpartitions modulo 4 and 8, Raman. J. 48 (2) (2019) 251
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.
Mirjana Vuletic, A generalization of MacMahon's formula, Trans. Am. Math. Soc. 361 (2009) 2789-2804.

Cf. A000219, A015128, A026007, A261384, A261386, A261389.
Cf. A206622, A206623, A206624, A261519, A261520.
Cf. A285462, A285447, A285460, A285461.
Cf. A285446, A285458, A285459.
Cf. A306081.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m,2)-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, May 01 2010

Convolve A000219 with A026007.
O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021

A046682 Number of cycle types of conjugacy classes of all even permutations of n elements.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 40, 52, 69, 90, 118, 151, 195, 248, 317, 400, 505, 632, 793, 985, 1224, 1512, 1867, 2291, 2811, 3431, 4186, 5084, 6168, 7456, 9005, 10836, 13026, 15613, 18692, 22316, 26613, 31659, 37619, 44601, 52815, 62416, 73680, 86809, 102162

Offset: 0

Vladeta Jovovic

Also number of partitions of n with even number of even parts. There is no restriction on the odd parts.
a(n) = u(n) + v(n), n >= 2, of the Osima reference, p. 383.
Also number of partitions of n with largest part congruent to n modulo 2: a(2*n) = A027187(2*n), a(2*n-1) = A027193(2*n-1); a(n) = A000041(n) - A000701(n). - Reinhard Zumkeller, Apr 22 2006
Equivalently, number of partitions of n with number of parts having the same parity as n. - Olivier Gérard, Apr 04 2012
Also number of distinct free Young diagrams (Ferrers graphs with n nodes). Free Young diagrams are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. - Jani Melik, May 08 2016
Let the cycle type of an even permutation be represented by the partition A=(O1,O2,...,Oi,E1,E2,...,E2j), where the Os are parts with odd length and the Es are parts with even lengths, and where j may be zero, using Reinhard Zumkeller's observation that the partition associated with a cycle type of an even permutation has an even number of even parts. The set of even cycle types enumerated here can be considered a monoid under a binary operation *: Let A be as above and B=(o1,o2,...,ok,e1,e2,...,e2m). A*B is the partition (O1o1,O1o2,...,O1ok,O1e1,...,O1e2m,O2o1,...,O2e2m,...,Oio1,...,Oie2m,E1o1,...,E1e2m,...,E2je2m). This product has 2im+2jk+4jm even parts, so it represents the cycle type of an even permutation. - Richard Locke Peterson, Aug 20 2018
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer partitions of n with Heinz number greater than or equal to that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352488. The complement is counted by A000701. For example, the a(n) partitions for n = 1...7 are:
  (1)  (11)  (21)   (22)    (221)    (222)     (331)
             (111)  (211)   (311)    (321)     (2221)
                    (1111)  (2111)   (2211)    (3211)
                            (11111)  (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)
Also the number of integer partitions of n with Heinz number less than or equal to their conjugate, ranked by A352489. For example, the a(n) partitions for n = 1...7 are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)
            (21)  (22)  (32)   (33)   (43)
                  (31)  (41)   (42)   (52)
                        (311)  (51)   (61)
                               (321)  (322)
                               (411)  (421)
                                      (511)
                                      (4111)
(End)

			1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + ...
a(3)=2 since cycle types of even permutations of 3 elements is (.)(.)(.), (...).
a(4)=3 since cycle types of even permutations of 4 elements is (.)(.)(.)(.), (...)(.), (..)(..).
a(5)=4 (free Young diagrams):
  XXXXX XXXX. XXX.. XXX..
  ..... X.... XX... X....
  ..... ..... ..... X....
  ..... ..... ..... .....
  ..... ..... ..... .....

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
J. Huh and B. Kim, The number of equivalence classes arising from partition involutions, Int. J. Number Theory, 16 (2020), 925-939.
M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
Sheila Sundaram, On a positivity conjecture in the character table of S_n, arXiv:1808.01416 [math.CO], 2018.

Cf. A000701, A006950, A015128.
For the number of conjugacy classes of the alternating group A_n, n>=2, see A000702.
Cf. A118301.
A000041 counts integer partitions.
A000700 counts self-conjugate partitions, ranked by A088902.
A330644 counts non-self-conjugate partitions, ranked by A352486.
Heinz number (rank) and partition:
- A122111 = rank of conjugate.
- A296150 = parts of partition, conjugate A321649.
- A352487 = rank less than conjugate, counted by A000701.
- A352488 = rank greater than or equal to conjugate, counted by A046682.
- A352489 = rank less than or equal to conjugate, counted by A046682.
- A352490 = rank greater than conjugate, counted by A000701.
- A352491 = rank minus conjugate.
Cf. A114088, A115994, A171966, A238352, A258116, A321648, A325039.

seq(add((-1)^(n-k)*combinat:-numbpart(n,k),k=0..n),n=0..48); # Peter Luschny, Aug 03 2015

max = 48; f[q_] := Sum[(-q^2)^n^2, {n, 0, max}]/Product[1-q^n, {n, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* Jean-François Alcover, Oct 18 2011, after g.f. *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>=Times@@Prime/@conj[#]&]],{n,0,15}] (* Gus Wiseman, Mar 31 2022 *)

list(lim)=my(q='q);Vec(sum(n=0,sqrt(lim),(-q^2)^(n^2))/prod(n=1,lim,1-q^n)+O(q^(lim\1+1))) \\ Charles R Greathouse IV, Oct 18 2011

{a(n) = if( n<0, 0, (numbpart(n) + polcoeff( 1 / prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)), n)) / 2)} /* Michael Somos, Jul 24 2012 */

G.f.: Sum_{n>=0} (-q^2)^(n^2) / Product_{m>=1} (1-q^m ) = ( 1/Product_{m>=1} (1-q^m) + Product_{m>=1} (1+q^(2*m-1) ) ) / 2. - Mamuka Jibladze, Sep 07 2003
a(n) = (A000041(n) + A000700(n)) / 2.
a(n) = A000041(n) - A000701(n). - Gus Wiseman, Mar 31 2022

A169975 Expansion of Product_{i>=0} (1 + x^(4*i+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 2, 6, 10, 6, 2, 6, 12, 9, 3, 7, 14, 11, 4, 7, 16, 15, 6, 8, 19, 18, 8, 9, 21, 23, 11, 10, 24, 27, 14, 11, 27, 34, 19, 13, 30, 39, 24, 15, 33, 47, 31, 18, 37, 54, 38

Offset: 0

N. J. A. Sloane, Aug 29 2010

nonn

Number of partitions into distinct parts of the form 4*k+1.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k + b)), then a(n) ~ exp(Pi*sqrt(n/(3*a))) / (2^(1 + b/a) * (3*a)^(1/4) * n^(3/4)) [Meinardus, 1954]. - Vaclav Kotesovec, Aug 26 2015
Convolution of A147599 and A169975 is A000700. - Vaclav Kotesovec, Jan 18 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Günter Meinardus, Über Partitionen mit Differenzenbedingungen, Mathematische Zeitschrift (1954/55), Volume: 61, page 289-302

Cf. A000700, A261612, A280454, A280456, A280457.
Cf. A015128, A080054, A261610, A261611.
Cf. A000041, A000009, A035382, A035451.
Cf. A170966-A170975.
Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 4] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Sum_{n>=0} (x^(2*n^2 - n) / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Mar 10 2011
G.f.: G(0)/x where G(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(4*k+1))/G(k+1) )); (recursively defined continued fraction, see A006950). - Sergei N. Gladkovskii, Jan 28 2013
a(n) ~ exp(Pi*sqrt(n)/(2*sqrt(3))) / (2^(7/4) * 3^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 18 2017

A001934 Expansion of 1/theta_4(q)^2 in powers of q.

Original entry on oeis.org

1, 4, 12, 32, 76, 168, 352, 704, 1356, 2532, 4600, 8160, 14176, 24168, 40512, 66880, 108876, 174984, 277932, 436640, 679032, 1046016, 1597088, 2418240, 3632992, 5417708, 8022840, 11802176, 17252928, 25070568, 36223424, 52053760, 74414412

Offset: 0

N. J. A. Sloane, Simon Plouffe

Euler transform of period 2 sequence [ 4, 2, ...].
The Cayley reference actually is to A004403. - Michael Somos, Feb 24 2011
Number of overpartition pairs, see Lovejoy reference. - _Joerg Arndt, Apr 03 2011
In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^m and m>=1, then a(n) ~ exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)). - Vaclav Kotesovec, Aug 17 2015

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
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).

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
B. Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
Jeremy Lovejoy, Overpartition pairs, Annales de l'institut Fourier, vol.56, no.3, p.781-794, 2006.

Cf. A000122, A000203, A004403, A002318, A002513, A015128, A004404-A004425.

# JacobiTheta4 is defined in A002448.
A001934List(len) = JacobiTheta4(len, -2)
A001934List(33) |> println # Peter Luschny, Mar 12 2018

Maple
```
mul((1+x^n)^2/(1-x^n)^2,n=1..256);
```

CoefficientList[Series[1/EllipticTheta[4, 0, q]^2, {q, 0, 32}], q]  (* Jean-François Alcover, Jul 18 2011 *)
nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
QP = QPochhammer; s = QP[q^2]^2/QP[q]^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

my(N=33, x='x+O('x^N)); Vec(prod(i=1, N, (1+x^i)^2/(1-x^i)^2))

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4, n))} /* Michael Somos, Feb 09 2006 */

G.f.: Product ( 1 - x^k )^{-c(k)}, c(k) = 4, 2, 4, 2, 4, 2, ....
G.f.: Product{i>=1} (1+x^i)^2/(1-x^i)^2. - Jon Perry, Apr 04 2004
Expansion of eta(q^2)^2/eta(q)^4 in powers of q, where eta(x)=prod(n>=1,1-q^n).
a(n) = (-1)^n * A004403(n). a(n) = 4 * A002318(n) unless n=0. - Michael Somos, Feb 24 2011
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(15/4) * n^(5/4)) * (1 - 15/(8*Pi*sqrt(2*n)) + 105/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 17 2015, extended Jan 22 2017
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
The g.f. A(q^2) = 1/(F(q)*F(-q)), where F(q) = theta_3(q) = Sum_{n = -oo..oo} q^(n^2) is the g.f. of A000122. Cf. A002513. - Peter Bala, Sep 26 2023

More terms from James Sellers, Sep 08 2000

Edited by N. J. A. Sloane, May 13 2008 to remove an incorrect g.f.

A261612 Expansion of Product_{k>=0} (1 + x^(3*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 3, 3, 2, 4, 4, 2, 4, 5, 3, 5, 7, 4, 5, 8, 6, 7, 10, 7, 7, 12, 10, 9, 14, 12, 10, 16, 16, 13, 19, 19, 15, 22, 24, 19, 25, 28, 22, 29, 35, 28, 33, 40, 33, 38, 48, 41, 44, 55, 48, 51, 66, 59, 58, 74, 69

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A000700, A035382, A169975.

Cf. A015128, A080054, A261610, A261611, A262928.

nmax = 100; CoefficientList[Series[Product[(1 + x^(3*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)

a(n) ~ exp(Pi*sqrt(n)/3) / (2^(4/3) * sqrt(3) * n^(3/4)) * (1 - (Pi/144 + 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 16 2017

G.f.: Sum_{k>=0} x^(k*(3*k - 1)/2) / Product_{j=1..k} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 24 2020

A002513 Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.

Original entry on oeis.org

1, 1, 3, 4, 9, 12, 23, 31, 54, 73, 118, 159, 246, 329, 489, 651, 940, 1242, 1751, 2298, 3177, 4142, 5630, 7293, 9776, 12584, 16659, 21320, 27922, 35532, 46092, 58342, 75039, 94503, 120615, 151173, 191611, 239060, 301086, 374026, 468342, 579408, 721638, 889287

Offset: 0

N. J. A. Sloane, Simon Plouffe

For a real polynomial equation of degree n, a(n) is the number of possibilities for the roots to be real and unequal, real and equal (in various combinations), or simple or multiple complex conjugates. For example, a(3)=4 because we can have: three equal roots, two equal roots, three distinct real roots and two complex roots (see the Monthly Problem reference). - Emeric Deutsch, Mar 22 2005
Number of partitions of n, the even parts being of two kinds. E.g. a(4)=9 because we have 4, 4', 3+1, 2+2, 2+2', 2'+2', 2+1+1, 2'+1+1, 1+1+1+1. - Emeric Deutsch, Mar 22 2005
For the name "cubic partition" see Xiong; Chen & Lin; Chern & Dastidar. - Michel Marcus, Jan 28 2016

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 23*x^6 + 31*x^7 + 54*x^8 + 73*x^9 + ...
G.f. = 1/q + q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 23*q^47 + 31*q^55 + 54*q^63 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0930 and N0931).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Zakir Ahmed, Nayandeep Deka Baruah, and Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Koustav Banerjee, New Asymptotics and Inequalities Related to the Partition Function, Doctoral Thesis, Johannes Kepler Univ. (Linz, Austria 2022).
M. F. Capobianco and C. F. Pinzka, Problem 2055, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
William Y. C. Chen and Bernard L. S. Lin, Congruences for the Number of Cubic Partitions Derived from Modular Forms, arXiv:0910.1263 [math.NT], 2016.
Shane Chern and Manosij Ghosh Dastidar, Congruences and recursions for the cubic partitions, arXiv:1601.06480 [math.NT], 2016.
Marston Conder, Tomaš Pisanski, and Arjana Žitnik, Vertex-transitive graphs and their arc-types, arXiv preprint arXiv:1505.02029 [math.CO], 2015.
R. K. Guy, Letter to Morris Newman, Aug 21 1986, concerning A2513 (annotated scanned copy, with permission).
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 16.
Vaclav Kotesovec, Asymptotics of sequence A002513, 2019.
Lukas Mauth, Exact formula for cubic partitions, arXiv:2305.03396 [math.NT], 2023.
Morris Newman, Construction and application of a class of modular functions (II). Proc. London Math. Soc. (3) 9 1959 373-387.
Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
James A. Sellers, Elementary proofs of congruences for the cubic and overcubic partition functions, Australasian Journal of Combinatorics, Volume 60(2) (2014), Pages 191-197.
Xinhua Xiong, The number of cubic partitions modulo powers of 5, arXiv:1004.4737 [math.NT], 2010.

Cf. A000122, A015128, A001934, A089799, A215947, A273225, A319455.

N:= 50: # to get a(0) to a(N)
P:= mul((1-x^(2*k))^(-2)*(1-x^(2*k-1))^(-1),k=1..ceil(N/2)):
S:= series(P, x, N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 26 2016
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(d::odd, d, 2*d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 05 2020

max = 50; f[x_] := Product[ 1/((1-x^(2 k))^2*(1-x^(2k-1))), {k, 1, Ceiling[max/2]} ]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 04 2011 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q] / QPochhammer[ q^2], {q, 0, n}];(* Michael Somos, Jul 17 2013 *)
Table[Sum[PartitionsP[k]*PartitionsP[n-2k],{k,0,n/2}],{n,0,50}] (* Vaclav Kotesovec, Jun 22 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A) / eta(x^2 + A), n))}; /* Michael Somos, Nov 10 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 2)
a = EulerTransform(b)
print([a(n) for n in range(44)]) # Peter Luschny, Nov 17 2022

From Michael Somos, Mar 23 2003: (Start)
Expansion of q^(1/8) / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 2 sequence [1, 2, ...].
G.f.: Product_{k>0} 1/((1 - x^(2*k))^2 * (1 - x^(2*k-1))).
(End)
Given g.f. A(x), then B(q) = A(q)^8 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 16*v^4 + v^3*w + 256*u*v^3 + 16*u*v^2*w - u^2*w^2. - Michael Somos, Apr 03 2005
a(n) ~ exp(Pi*sqrt(n)) / (8*n^(5/4)) * (1 - (Pi/16 + 15/(8*Pi))/sqrt(n)). - Vaclav Kotesovec, Jun 22 2015, extended Jan 17 2017
From Michel Marcus, Jan 28 2016: (Start)
G.f.: Product_{k>0} 1/((1 - x^k) * (1 - x^(2*k))).
a(3n+2) = 0 (mod 3).
a(25n+22) = 0 (mod 5) (see Xiong).
a(49n+15) = a(49n+29) = a(49n+36) = a(49n+43) = 0 (mod 7) (see Chen & Lin).
a(297n+62) = a(297n+161) = 0 (mod 11) (see Chern & Dastidar).
(End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(-7/2) (t/i)^-1 f(t) where q = exp(2 Pi i t). - Michael Somos, Oct 17 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018
From Peter Bala, Sep 25 2023: (Start)
The g.f. A(x) satisfies log(A(x)) = x + 5*x^2/2 + 4*x^3/3 + 13*x^4/4 + ... = Sum_{n >= 1} A215947(n)*x^n/n.
A(x^2) = 4/(F(x)*F(-x)) = 2/(F(x)*G(-x)), where F(x) = Sum_{n = -oo..oo} x^(n*(n+1)/2) is the g.f. of A089799 and G(x) = Sum_{n = -oo..oo} x^(n^2) is the g.f. of A000122. Cf. A001934. Note that 4/(F(-x)*F(-x)) is the g.f. of A273225.
The self-convolution A(x)^2 is the g.f. of A319455. (End)

More terms and information from Michael Somos, Mar 23 2003

cp's OEIS Frontend

A356290 a(n) = Sum_{k=0..n} binomial(3*n, n-k) * v(k), where v(k) is the number of overpartitions of n (A015128).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Julia

Magma

Maple

Mathematica

Maxima

MuPAD

PARI

PARI

PARI

PARI

PARI

PARI

Perl

Python

Racket

Sage

Sage

Sage

Formula

Extensions

A006950 G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A067661 Number of partitions of n into distinct parts such that number of parts is even.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A356290 a(n) = Sum_{k=0..n} binomial(3n, n-k) v(k), where v(k) is the number of overpartitions of n (A015128).

A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).