A171966 -id:A171966 - cp's OEIS frontend

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

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

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

Original entry on oeis.org

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are Heinz numbers of the integer partitions counted by A045931.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - Peter Munn, Oct 05 2020

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   51: {2,7}
   58: {1,10}
   65: {3,6}
   69: {2,9}
   74: {1,12}
   77: {4,5}
   84: {1,1,2,4}
   86: {1,14}
   90: {1,2,2,3}
   93: {2,11}
   95: {3,8}

David A. Corneth, Table of n, a(n) for n = 1..10000

Positions of 0's in A195017.
A257992(n) = A257991(n).
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Closed under: A003961, A003991, A297845.
Subsequence of A028260, A332820.

Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]

is(n) = {my(v = vector(2), f = factor(n));for(i = 1, #f~,v[1 + primepi(f[i, 1])%2]+=f[i, 2]);v[1] == v[2]} \\ David A. Corneth, Oct 06 2020

from sympy import factorint, primepi
def ok(n):
    v = [0, 0]
    for p, e in factorint(n).items(): v[primepi(p)%2] += e
    return v[0] == v[1]
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Apr 16 2022 after David A. Corneth

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

David W. Wilson

nonn

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

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..3500 (first 1001 terms from David W. Wilson)

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.

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

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 *)

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

A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 190, 242, 310, 392, 497, 623, 782, 973, 1212, 1498, 1851, 2274, 2793, 3411, 4163, 5059, 6142, 7427, 8972, 10801, 12989, 15572, 18646, 22267, 26561, 31602, 37556, 44533, 52743, 62338, 73593

Offset: 0

N. J. A. Sloane

Also number of cycle types of odd permutations.
Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - N. Sato, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch, Mar 02 2006
Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller, Apr 22 2006
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer partitions of n with Heinz number greater than 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 A352490. The complement is counted by A046682. For example, the a(n) partitions for n = 2...8 are:
  (11)  (111)  (211)   (221)    (222)     (331)      (2222)
               (1111)  (2111)   (2211)    (2221)     (3221)
                       (11111)  (3111)    (3211)     (3311)
                                (21111)   (22111)    (22211)
                                (111111)  (31111)    (32111)
                                          (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
Also the number of integer partitions of n with Heinz number less than that of their conjugate, ranked by A352487. For example, the a(n) partitions for n = 2...8 are:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (31)  (32)  (33)   (43)   (44)
                  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (411)  (322)  (71)
                               (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (5111)
(End)

			G.f. = x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Brian Hopkins and Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80, see p. 10.
M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

Cf. A000041, A027187, A027193, A046682, A118302, A236559, A236914.
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, A325039.

with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;

a41 = PartitionsP; a700[n_] := SeriesCoefficient[ Product[1 + x^k, {k, 1, n, 2}], {x, 0, n}]; a[0] = 0; a[n_] := (a41[n] - a700[n])/2; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Feb 21 2012, after first formula *)
a[ n_] := SeriesCoefficient[ (1 / QPochhammer[ x] - 1 / QPochhammer[ x, -x]) / 2, {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x^2]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] Sum[ x^(2 k) / QPochhammer[ x^2, x^2, k], {k, 1, n/2, 2}], {x, 0, n}] (* Michael Somos, Aug 25 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (1 / QPochhammer[ x, x, k]^2 - 1 / QPochhammer[ x^2, x^2, k]) x^k^2, {k, Sqrt@n}] / 2, {x, 0, n}]]; (* Michael Somos, Aug 25 2015 *)
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 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x^2 + A)^2 / eta(x^4 + A) ) / (2 * eta(x + A)), n))}; /* Michael Somos, Aug 25 2015 */

q='q+O('q^60); concat([0, 0], Vec((1-eta(q^2)^2/eta(q^4))/(2*eta(q)))) \\ Altug Alkan, Sep 26 2018

a(n) = (A000041(n) - A000700(n))/2.
From Bill Gosper, Aug 08 2005: (Start)
Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...
= -( Sum_{n>=1} (-q^2)^(n^2) ) / ( Sum_{ n = -oo..oo } (-1)^n q^(n(3n-1)/2) )
= (- q; q){oo} Sum{n>=1} q^(2(2n-1))/(q^2;q^2)_{2n-1}
= (1/(q;q)_oo - 1/(q;-q)_oo)/2
= (1/(q;q)_oo - (-q;q^2)_oo)/2
= Sum{k>=0} ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2
using the "q-Pochhammer" notation (a;q)n := Product{k=0..n-1} (1 - a*q^k).
(End)
a(n) = p(n-2) - p(n-8) + p(n-18) - p(n-32) + ... + (-1)^(k+1)*p(n-2*k^2) + ..., where p() is A000041(). E.g., a(20) = p(18) - p(12) + p(2) = 385 - 77 + 2 = 310. - Vladeta Jovovic, Aug 08 2004
G.f.: (1/2)*(1 - Product_{j>=1} (1-x^(2j))/(1+x^(2j)))/Product_{j>=1} (1 - x^j). - Emeric Deutsch, Mar 02 2006
a(2*n) = A236559(n). a(2*n + 1) = A236914(n). - Michael Somos, Aug 25 2015
a(n) = A330644(n)/2. - Omar E. Pol, Jan 10 2020
a(n) = A000041(n) - A046682(n) = A046682(n) - A000700(n). - Gus Wiseman, Mar 31 2022

Better description and more terms from Christian G. Bower, Apr 27 2000

A063886 Number of n-step walks on a line starting from the origin but not returning to it.

Original entry on oeis.org

1, 2, 2, 4, 6, 12, 20, 40, 70, 140, 252, 504, 924, 1848, 3432, 6864, 12870, 25740, 48620, 97240, 184756, 369512, 705432, 1410864, 2704156, 5408312, 10400600, 20801200, 40116600, 80233200, 155117520, 310235040, 601080390, 1202160780, 2333606220, 4667212440

Offset: 0

Henry Bottomley, Aug 28 2001

A Chebyshev transform of A007877(n+1). The g.f. is transformed to (1+x)/((1-x)(1+x^2)) under the mapping G(x)->(1/(1+x^2))G(1/(1+x^2)). - Paul Barry, Oct 12 2004
a(n-1) = 2*C(n-2, floor((n-2)/2)) is also the number of bit strings of length n in which the number of 00 substrings is equal to the number of 11 substrings. For example, when n = 4 we have 4 such bit strings: 0011, 0101, 1010, and 1100. - Angel Plaza, Apr 23 2009
Hankel transform is A120617. - Paul Barry, Aug 10 2009
The Hankel transform of a(n) is (-2)^C(n+1,2). The Hankel transform of (-1)^C(n+1,2)*a(n) is (-1)^C(n+1,2)*A164584(n). - Paul Barry, Aug 17 2009
For n > 1, a(n) is also the number of n-step walks starting from the origin and returning to it exactly once. - Geoffrey Critzer, Jan 24 2010
-a(n) is the Z-sequence for the Riordan array A130777. (See the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices). - Wolfdieter Lang, Jul 12 2011
Number of subsets of {1,...,n} in which the even elements appear as often at even positions as at odd positions. - Gus Wiseman, Mar 17 2018

			a(4) = 6 because there are six length four walks that do not return to the origin: {-1, -2, -3, -4}, {-1, -2, -3, -2}, {-1, -2, -1, -2}, {1, 2, 1, 2}, {1, 2, 3, 2}, {1, 2, 3, 4}. There are also six such walks that return exactly one time: {-1, -2, -1, 0}, {-1, 0, -1, -2}, {-1, 0, 1, 2}, {1, 0, -1, -2}, {1, 0, 1, 2}, {1, 2, 1, 0}. - _Geoffrey Critzer_, Jan 24 2010
The a(5) = 12 subsets in which the even elements appear as often at even positions as at odd positions: {}, {1}, {3}, {5}, {1,3}, {1,5}, {2,4}, {3,5}, {1,2,4}, {1,3,5}, {2,4,5}, {1,2,4,5}. - _Gus Wiseman_, Mar 17 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, Problem 11424, The American Mathematical Monthly, Vol. 116, No. 3 (March 2009), p. 277.
D. Perrin, A conjecture on rational sequences, pp. 267-274 of R. M. Capocelli, ed., Sequences, Springer-Verlag, NY 1990.

Apart from initial terms, same as A182027.
Cf. A000108, A000246, A000712, A000984, A001405, A001700, A002420, A006232, A007877, A026010, A028329, A045931, A047073, A089849, A097613, A106180, A120617, A130777, A130780, A138364, A164584, A171966, A239241, A300787, A300788, A300789.
Cf. A307768 (complementary event).

[1] cat [2*Binomial(n-1, Floor((n-1)/2)): n in [1..40]]; // G. C. Greubel, Jun 07 2023

seq(seq(binomial(2*j,j)*i, i=1..2),j=0..16); # Zerinvary Lajos, Apr 28 2007
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
       4*a(n-2) +2*(a(n-1) -4*a(n-2))/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2014
# third program:
A063886 := series(BesselI(0, 2*x)*(1 + x*2 + x*Pi*StruveL(1, 2*x)) - Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x), x = 0, 34): seq(n!*coeff(A063886, x, n), n = 0 .. 33); # Mélika Tebni, Jun 17 2024

Table[Length[Select[Map[Accumulate, Strings[{-1, 1}, n]], Count[ #, 0] == 0 &]], {n, 0, 20}] (* Geoffrey Critzer, Jan 24 2010 *)
CoefficientList[Series[Sqrt[(1+2x)/(1-2x)],{x,0,40}],x] (* Harvey P. Dale, Apr 28 2016 *)

PARI
```
a(n)=(n==0)+2*binomial(n-1,(n-1)\2)
```

a(n) = 2^n*prod(k=0,n-1,(k/n+1/n)^((-1)^k)); \\ Michel Marcus, Dec 03 2013

from math import ceil
from sympy import binomial
def a(n):
    if n==0: return 1
    return 2*binomial(n-1,(n-1)//2)
print([a(n) for n in range(18)])
# David Nacin, Feb 29 2012

[2*binomial(n-1, (n-1)//2) + int(n==0) for n in range(41)] # G. C. Greubel, Jun 07 2023

G.f.: sqrt((1+2*x)/(1-2*x)).
a(n+1) = 2*C(n, floor(n/2)) = 2*A001405(n); a(2n) = C(2n, n) = A000984(n) = 4*a(2n-2)-|A002420(n)| = 4*a(2n-2)-2*A000108(n-1) = 2*A001700(n-1); a(2n+1) = 2*a(2n) = A028329(n).
2*a(n) = A047073(n+1).
a(n) = Sum_{k=0..n} abs(A106180(n,k)). - Philippe Deléham, Oct 06 2006
a(n) = Sum_{k=0..n} (k+1)binomial(n, (n-k)/2) ( 1-cos((k+1)*Pi/2) (1+(-1)^(n-k))/(n+k+2) ). - Paul Barry, Oct 12 2004
G.f.: 1/(1-2*x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+ ... (continued fraction). - Paul Barry, Aug 10 2009
G.f.: 1 + 2*x/(G(0)-x+x^2) where G(k)= 1 - 2*x^2 - x^4/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 10 2012
D-finite with recurrence: n*a(n) = 2*a(n-1) + 4*(n-2)*a(n-2). - R. J. Mathar, Dec 03 2012
From Sergei N. Gladkovskii, Jul 26 2013: (Start)
G.f.: 1/G(0), where G(k) = 1 - 2*x/(1 + 2*x/(1 + 1/G(k+1) )); (continued fraction).
G.f.: G(0), where G(k) = 1 + 2*x/(1 - 2*x/(1 + 1/G(k+1) )); (continued fraction).
G.f.: W(0)/2*(1+2*x), where W(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/W(k+1) )), abs(x) < 1/2; (continued fraction). (End)
a(n) = 2^n*Product_{k=0..n-1} (k/n + 1/n)^((-1)^k). - Peter Luschny, Dec 02 2013
G.f.: G(0), where G(k) = 1 + 2*x*(4*k+1)/((2*k+1)*(1+2*x) - (2*k+1)*(4*k+3)*x*(1+2*x)/((4*k+3)*x + (k+1)*(1+2*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014
From Peter Bala, Mar 29 2024: (Start)
a(n) = 2^n * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(- 1/2, n-k) = 2^n * A000246(n)/n!.
a(n) = (1/2^n) * binomial(2*n, n) * hypergeom([-1/2, -n], [1/2 - n], -1). (End)
E.g.f.: BesselI(0, 2*x)*(1 + x*(2 + Pi)*StruveL(1, 2*x)) - Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x). - Stefano Spezia, May 11 2024
a(n) = A089849(n) + A138364(n). - Mélika Tebni, Jun 17 2024
From Amiram Eldar, Aug 15 2025: (Start)
Sum_{n>=0} 1/a(n) = Pi/(3*sqrt(3)) + 2.
Sum_{n>=0} (-1)^n/a(n) = 2/3 + Pi/(9*sqrt(3)). (End)

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

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

Vladeta Jovovic, Aug 19 2007

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

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A045931, A108949, A108950.

Cf. A171966, A171967. - Reinhard Zumkeller, Jan 21 2010

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

$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 *)

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

More terms from Emeric Deutsch, Aug 24 2007

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2

Offset: 1

Gus Wiseman, Oct 22 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239(n).

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366531(n).

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

Alois P. Heinz, Mar 30 2014

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).

			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;

Alois P. Heinz, Rows n = 0..120, flattened

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.

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);

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 *)

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 */

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

Gus Wiseman, Jan 21 2022

nonn

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.

			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)

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.

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]&]

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

cp's OEIS Frontend

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

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A063886 Number of n-step walks on a line starting from the origin but not returning to it.

Views

Author

Keywords

Comments

Examples

Links