This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000041 M0663 N0244 #1145 Aug 07 2025 08:36:23
%S A000041 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,
%T A000041 1002,1255,1575,1958,2436,3010,3718,4565,5604,6842,8349,10143,12310,
%U A000041 14883,17977,21637,26015,31185,37338,44583,53174,63261,75175,89134,105558,124754,147273,173525
%N A000041 a(n) is the number of partitions of n (the partition numbers).
%C A000041 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
%C A000041 a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).
%C A000041 Also the number of rooted trees with n+1 nodes and height at most 2.
%C A000041 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
%C A000041 Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). - _Lekraj Beedassy_, Oct 16 2004
%C A000041 Number of graphs on n vertices that do not contain P3 as an induced subgraph. - _Washington Bomfim_, May 10 2005
%C A000041 Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - _Thomas Baruchel_, Nov 07 2005
%C A000041 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
%C A000041 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
%C A000041 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
%C A000041 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
%C A000041 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
%C A000041 From _Jerome Malenfant_, Feb 14 2011: (Start)
%C A000041 The matrix of a(n) values
%C A000041 a(0)
%C A000041 a(1) a(0)
%C A000041 a(2) a(1) a(0)
%C A000041 a(3) a(2) a(1) a(0)
%C A000041 ....
%C A000041 a(n) a(n-1) a(n-2) ... a(0)
%C A000041 is the inverse of the matrix
%C A000041 1
%C A000041 -1 1
%C A000041 -1 -1 1
%C A000041 0 -1 -1 1
%C A000041 ....
%C A000041 -d_n -d_(n-1) -d_(n-2) ... -d_1 1
%C A000041 where d_q = (-1)^(m+1) if q = m(3m-1)/2 = the m-th generalized pentagonal number (A001318), = 0 otherwise. (End)
%C A000041 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
%C A000041 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
%C A000041 a(n) is number of distinct characteristic polynomials among n! of permutations matrices size n X n. - _Artur Jasinski_, Oct 24 2011
%C A000041 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)
%C A000041 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
%C A000041 Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - _Zhi-Wei Sun_, Dec 02 2013
%C A000041 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
%C A000041 a(n) is the number of compositions of n into positive parts avoiding the pattern [1, 2]. - _Bob Selcoe_, Jul 08 2014
%C A000041 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
%C A000041 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
%C A000041 Define a segmented partition a(n,k, <s(1)..s(j)>) 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
%C A000041 (End)
%C A000041 From _Gregory L. Simay_, Nov 09 2015: (Start)
%C A000041 The polynomials for a(n, k, <s(1), ..., s(j)>) have degree j-1.
%C A000041 a(n, k, <k>) = 1 if n = 0 mod k, = 0 otherwise
%C A000041 a(rn, rk, <r*s(1), ..., r*s(j)>) = a(n, k, <s(1), ..., s(j)>)
%C A000041 a(n odd, k, <all s(j) even>) = 0
%C A000041 Established results can be recast in terms of segmented partitions:
%C A000041 For j(j+1)/2 <= n < (j+1)(j+2)/2, A000009(n) = a(n, 1, <1>) + ... + a(n, j, <j 1's>), j < n
%C A000041 a(n, k, <j 1's>) = a(n - j(j-1)/2, k)
%C A000041 (End)
%C A000041 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
%C A000041 Satisfies Benford's law [Anderson-Rolen-Stoehr, 2011]. - _N. J. A. Sloane_, Feb 08 2017
%C A000041 The partition function p(n) is log-concave for all n>25 [DeSalvo-Pak, 2014]. - _Michel Marcus_, Apr 30 2019
%C A000041 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
%C A000041 Number of equivalence relations on n unlabeled nodes. - _Lorenzo Sauras Altuzarra_, Jun 13 2022
%C A000041 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
%C A000041 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
%C A000041 a(n) is also the number of partitions of n*(n+3)/2 into n distinct parts. - _David García Herrero_, Aug 20 2024
%C A000041 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
%D A000041 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
%D A000041 George E. Andrews and K. Ericksson, Integer Partitions, Cambridge University Press 2004.
%D A000041 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 307.
%D A000041 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III.
%D A000041 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
%D A000041 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.
%D A000041 Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag.
%D A000041 B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.
%D A000041 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 999.
%D A000041 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.
%D A000041 Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 411.
%D A000041 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 94-96.
%D A000041 L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp. 101-164, Chelsea NY 1992.
%D A000041 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 37, Eq. (22.13).
%D A000041 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000041 G. H. Hardy and S. Ramanujan, Asymptotic formulas in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-.
%D A000041 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 83-100, 113-131.
%D A000041 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, 273-296.
%D A000041 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 396.
%D A000041 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.1, p. 491.
%D A000041 S. Ramanujan, Collected Papers, Chap. 25, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1919), pp. 207-213).
%D A000041 S. Ramanujan, Collected Papers, Chap. 28, Cambridge Univ. Press 1927 (Proceedings of the London Math. Soc., 2, 18(1920)).
%D A000041 S. Ramanujan, Collected Papers, Chap. 30, Cambridge Univ. Press 1927 (Mathematische Zeitschrift, 9 (1921), pp. 147-163).
%D A000041 S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Table IV on page 308.
%D A000041 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 122.
%D A000041 J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.
%D A000041 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000041 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000041 R. E. Tapscott and D. Marcovich, "Enumeration of Permutational Isomers: The Porphyrins", Journal of Chemical Education, 55 (1978), 446-447.
%D A000041 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 286-289, 297-298, 303.
%D A000041 Robert M. Young, "Excursions in Calculus", Mathematical Association of America, p. 367.
%H A000041 David W. Wilson, <a href="/A000041/b000041.txt">Table of n, a(n) for n = 0..10000</a>
%H A000041 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 836. [scanned copy]
%H A000041 Scott Ahlgren and Ken Ono, <a href="http://www.ams.org/notices/200109/fea-ahlgren.pdf">Addition and Counting: The Arithmetic of Partitions</a>, Notices of the AMS, 48 (2001) pp. 978-984.
%H A000041 Scott Ahlgren and Ken Ono, <a href="http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=60793">Congruence properties for the partition function</a>
%H A000041 Scott Ahlgren and Ken Ono, <a href="http://dx.doi.org/10.1073/pnas.191488598">Congruence properties for the partition function</a>, PNAS, vol. 98 no. 23, 12882-12884.
%H A000041 Gert Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H A000041 Gert Almkvist, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6126.pdf">On the differences of the partition function</a>, Acta Arith., 61.2 (1992), 173-181.
%H A000041 Gert Almkvist and Herbert S. Wilf, <a href="http://citeseer.nj.nec.com/correct/513487">On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n)</a>. [Broken link?]
%H A000041 Gert Almkvist and Herbert S. Wilf, <a href="https://doi.org/10.1006/jnth.1995.1027">On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n)</a>, Journal of Number Theory, Vol. 50, No. 2, 1995, pp. 329-334.
%H A000041 Amazing Mathematical Object Factory, <a href="https://web.archive.org/web/20070920114320/http://www.aarms.math.ca/ACMN/amof/e_partI.htm">Information on Partitions</a>. [Wayback Machine link]
%H A000041 Edward Anderson, <a href="https://arxiv.org/abs/1805.03346">Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces</a>, arXiv:1805.03346 [gr-qc], 2018.
%H A000041 Theresa C. Anderson, Larry Rolen and Ruth Stoehr, <a href="https://doi.org/10.1090/S0002-9939-2010-10577-4">Benford's Law for Coefficients of Modular Forms and Partition Functions</a>, Proceedings of the American Mathematical Society, Vol. 139, No. 5, May 2011, pp. 1533-1541.
%H A000041 George E. Andrews, <a href="http://www.emis.de/journals/SLC/opapers/s25andrews.html">Three Aspects of Partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%H A000041 George E. Andrews, <a href="https://doi.org/10.37236/1858">On a Partition Function of Richard Stanley</a>, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R1.
%H A000041 George E. Andrews and Ken Ono, <a href="http://pubmedcentral.com/articlerender.fcgi?artid=1266147">Ramanujan's congruences and Dyson's crank</a>
%H A000041 George E. Andrews and Ranjan Roy, <a href="https://doi.org/10.37236/1317">Ramanujan's Method in q-series Congruences</a>, The Electronic Journal of Combinatorics, Volume 4, Issue 2 (1997) (The Wilf Festschrift volume) > Research Paper #R2.
%H A000041 George E. Andrews, Sumit Kumar Jha, and J. López-Bonilla, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Jha/jha25.pdf">Sums of Squares, Triangular Numbers, and Divisor Sums</a>, Journal of Integer Sequences, Vol. 26 (2023), Article 23.2.5.
%H A000041 Anonymous, <a href="http://felix.unife.it/Root/d-Mathematics/d-Number-theory/b-Partitions">Bibliography on Partitions</a>
%H A000041 Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://doi.org/10.1007/s10801-024-01318-x">An ultimately periodic chain in the integral Lie ring of partitions</a>, J. Algebr. Comb. (2024). See p. 11.
%H A000041 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4, pp.344-353.
%H A000041 A. O. L. Atkins and F. G. Garvan, <a href="https://arxiv.org/abs/math/0208050">Relations between the ranks and cranks of partitions</a>, arXiv:math/0208050 [math.NT], 2002.
%H A000041 Helena Bergold, Lukas Egeling, and Hung. P. Hoang, <a href="https://arxiv.org/abs/2411.19208">Signotopes with few plus signs</a>, arXiv:2411.19208 [math.CO], 2024. See p. 14.
%H A000041 Alexander Berkovich and Frank G. Garvan, <a href="https://arxiv.org/abs/math/0401012">On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5</a>, arXiv:math/0401012 [math.CO], 2004.
%H A000041 Alexander Berkovich and Frank G. Garvan, <a href="https://arxiv.org/abs/math/0402439">On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations</a>, arXiv:math/0402439 [math.CO], 2004.
%H A000041 Bruce C. Berndt, <a href="http://www.math.uiuc.edu/~berndt/articles/partitions2.pdf">Ramanujan's congruences for the partition function modulo 5,7 and 11</a>
%H A000041 Bruce C. Berndt and K. Ono, <a href="http://www.math.wisc.edu/~ono/reprints/044.pdf">Ramanujan's Unpublished Manuscript On The Partition And Tau Functions With Proofs And Commentary</a>
%H A000041 Bruce C. Berndt and K. Ono, <a href="http://emis.dsd.sztaki.hu/journals/SLC/wpapers/s42berndt.html">Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary</a>, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
%H A000041 Frits Beukers, <a href="https://www.pyth.eu/uploads/user/ArchiefPDF/Pyth38-6.pdf">Ramanujan and The Partition Function (text in Dutch)</a>, Pythagoras, Wiskundetijdschrift voor Jongeren, 38ste Jaargang, Nummer 6, Agustus 1999, pp. 15-16.
%H A000041 Henry Bottomley, <a href="/A008284/a008284.gif">Illustration of initial terms</a>
%H A000041 Henry Bottomley, <a href="/A000009/a000009.gif">Illustration of initial terms of A000009, A000041 and A047967</a>
%H A000041 Henry Bottomley, <a href="http://www.btinternet.com/~se16/js/partitions.htm">Partition and composition calculator</a> [broken link]
%H A000041 Kevin S. Brown, <a href="http://www.math.niu.edu/~rusin/known-math/95/partitions">Additive Partitions of Numbers</a> [Broken link]
%H A000041 Kevin S. Brown, <a href="/A000041/a000041_1.txt">Additive Partitions of Numbers</a> [Cached copy of lost web page]
%H A000041 Kevin S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath383.htm">Computing the Partitions of n</a>
%H A000041 Jan Hendrik Bruinier, Amanda Folsom, Zachary A. Kent and Ken Ono, <a href="http://www.mathematik.tu-darmstadt.de/fbereiche/AlgGeoFA/staff/bruinier/publications/ramapofn125.pdf">Recent work on the partition function</a>
%H A000041 Jan Hendrik Bruinier and Ken Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/134.pdf">Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms</a>
%H A000041 Peter J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000041 Huantian Cao, <a href="http://cobweb.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002.
%H A000041 Huantian Cao, <a href="/A000009/a000009.pdf">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002 [Local copy, with permission]
%H A000041 Chao-Ping Chen and Hui-Jie Zhang, <a href="https://doi.org/10.1186/s13660-017-1479-8">Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality</a>, Journal of Inequalities and Applications, 2017.
%H A000041 Yuriy Choliy and Andrew V. Sills, <a href="http://home.dimacs.rutgers.edu/~asills/Durfee/CholiySillsRevAOC.pdf">A formula for the partition function that 'counts'</a>
%H A000041 Lynn Chua and Krishanu Roy Sankar, <a href="https://doi.org/10.37236/3675">Equipopularity Classes of 132-Avoiding Permutations</a>, 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
%H A000041 CombOS - Combinatorial Object Server, <a href="http://combos.org/part.html">Generate integer partitions</a>
%H A000041 Jimena Davis and Elizabeth Perez, <a href="http://www.ces.clemson.edu/~kevja/REU/2002/JDavisAndEPerez.pdf">Computations Of The Partition Function, p(n)</a>
%H A000041 Stephen DeSalvo and Igor Pak, <a href="https://arxiv.org/abs/1310.7982">Log-Concavity of the Partition Function</a>, arXiv:1310.7982 [math.CO], 2013-2014.
%H A000041 F. J. Dyson, <a href="https://archim.org.uk/eureka/archive/Eureka-8.pdf">Some guesses in the theory of partitions</a>, Eureka (Cambridge) 8 (1944), 10-15.
%H A000041 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1403.5664">Automatic Proofs of Asymptotic Abnormality (and much more!) of Natural Statistics Defined on Catalan-Counted Combinatorial Families</a>, arXiv:1403.5664 [math.CO], 2014.
%H A000041 Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, <a href="https://arxiv.org/abs/2004.08901">Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions</a>, arXiv:2004.08901 [math.CO], 2020.
%H A000041 FindStat - Combinatorial Statistic Finder, <a href="https://www.findstat.org/CollectionsDatabase/IntegerPartitions/">Integer partitions</a>
%H A000041 Nathan J. Fine, <a href="http://www.pnas.org/cgi/reprint/34/12/616.pdf">Some New Results On Partitions</a>
%H A000041 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 41.
%H A000041 Amanda Folsom, Zachary A. Kent and Ken Ono, <a href="http://www.aimath.org/news/partition/folsom-kent-ono.pdf">l-adic properties of the partition function</a>, in press.
%H A000041 Amanda Folsom, Zachary A. Kent and Ken Ono, <a href="http://dx.doi.org/10.1016/j.aim.2011.11.013">l-adic properties of the partition function</a>, Adv. Math. 229 (2012) 1586.
%H A000041 B. Forslund, <a href="http://my.tbaytel.net/~forslund/partitio.html">Partitioning Integers</a>
%H A000041 Harald Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/k1partn.html">Partitions of an Integer</a>
%H A000041 Bert Fristedt, <a href="https://doi.org/10.1090/S0002-9947-1993-1094553-1">The structure of random partitions of large integers</a>, Transactions of the American Mathematical Society, 337.2 (1993): 703-735. [A classic paper - _N. J. A. Sloane_, Aug 27 2018]
%H A000041 GEO magazine, <a href="http://www.geo.de/GEO/wissenschaft_natur/technik/2000_11_GEO_11_zahlenspalterei/">Zahlenspalterei</a>
%H A000041 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=NjCIq58rZ8I">Partitions</a>, Numberphile video (2016).
%H A000041 Harald Grosse, Alexander Hock, and Raimar Wulkenhaar, <a href="https://arxiv.org/abs/1903.12526">A Laplacian to compute intersection numbers on M_(g,n) and correlation functions in NCQFT</a>, arXiv:1903.12526 [math-ph], 2019.
%H A000041 G. H. Hardy and S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram36.pdf">Asymptotic formulas in combinatorial analysis</a>, Proc. London Math. Soc., 17 (1918), 75-115.
%H A000041 A. Hassen and T. J. Olsen, <a href="http://www.math.temple.edu/~melkamu/html/partition.pdf">Playing With Partitions On The Computer</a>
%H A000041 Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
%H A000041 Alexander D. Healy, <a href="http://www.alexhealy.net/papers/math192.pdf">Partition Identities</a>
%H A000041 Ferdinand Ihringer, <a href="https://arxiv.org/abs/2002.06601">Remarks on the Erdős Matching Conjecture for Vector Spaces</a>, arXiv:2002.06601 [math.CO], 2020.
%H A000041 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=61">Encyclopedia of Combinatorial Structures 61</a> and <a href="http://ecs.inria.fr/services/structure?nbr=74">Encyclopedia of Combinatorial Structures 74</a>
%H A000041 Fredrik Johansson, <a href="http://fredrikj.net/math/nist.pdf">Fast arbitrary-precision evaluation of special functions in the Arb library</a>, OPSFA13, NIST, June 2015, page 15.
%H A000041 Jonthan M. Kane, <a href="http://www.jstor.org/stable/2690190">Distribution of orders of Abelian groups</a>, Math. Mag., 49 (1976), 132-135.
%H A000041 Jerome Kelleher and Barry O'Sullivan, <a href="http://arxiv.org/abs/0909.2331">Generating All Partitions: A Comparison Of Two Encodings</a>, arXiv:0909.2331 [cs.DS], 2009-2014.
%H A000041 Erica Klarreich, <a href="http://www.sciencenews.org/articles/20050618/bob9.asp">Pieces of Numbers: A proof brings closure to a dramatic tale of partitions and primes</a>, Science News, Week of Jun 18, 2005; Vol. 167, No. 25, p. 392.
%H A000041 Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
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%H A000041 J. Laurendi, <a href="http://www.artofproblemsolving.com/Resources/Papers/LaurendiPartitions.pdf">Partitions of Integers</a>
%H A000041 Oleg Lazarev, Matt Mizuhara, and Ben Reid, <a href="http://www.math.oregonstate.edu/~swisherh/LazarevMizuharaReid.pdf">Some Results in Partitions, Plane Partitions, and Multipartitions</a>
%H A000041 Li Wenwei, <a href="http://arxiv.org/abs/1612.05526">Estimation of the Partition Number: After Hardy and Ramanujan</a>, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
%H A000041 T. Lockette, Explore Magazine, <a href="http://rgp.ufl.edu/explore/v05n2/math.html">Path To Partitions</a>
%H A000041 Jerome Malenfant, <a href="http://arxiv.org/abs/1103.1585">Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers</a>, arXiv:1103.1585 [math.NT], 2011.
%H A000041 Dr. Math, <a href="http://mathforum.org/dr.math/problems/partitions.html">Partitioning the Integers</a> and <a href="http://mathforum.org/dr.math/problems/huckin11.14.98.html">Partitioning an Integer</a>
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%H A000041 OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
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%H A000041 Ken Ono (with J. Bruinier, A. Folsom and Z. Kent), Emory University, <a href="http://www.youtube.com/watch?v=aj4FozCSg8g">Adding and counting</a>
%H A000041 T. J. Osler, <a href="http://www2.rowan.edu/mars/depts/math/HASSEN/NT/Playpart.html">Playing with Partitions on the Computer</a>
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%H A000041 Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper25/page1.htm">Some Properties Of p(n), The Number Of Partitions Of n</a>
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%H A000041 Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper30/page1.htm">Congruence Properties Of Partitions</a>
%H A000041 Srinivasa Ramanujan and G. H. Hardy, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper31/page1.htm">Une formule asymptotique pour le nombre de partitions de n</a>
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%H A000041 J. D. Rosenhouse, <a href="http://www.math.ksu.edu/~jasonr/Solutions4.pdf">Solutions to Problems</a>
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%H A000041 F. Ruskey, <a href="http://combos.org/part">Generate Numerical Partitions</a>
%H A000041 F. Ruskey, <a href="https://web.archive.org/web/20160604023632/http://www.theory.cs.uvic.ca/tables/partitions.txt.gz">The first 284547 partition numbers</a> (52MB compressed file, archived link)
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%H A000041 Zhumagali Shomanov, <a href="http://arxiv.org/abs/1508.03173">Combinatorial formula for the partition function</a>, arXiv:1508.03173 [math.CO], 2015.
%H A000041 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series011">Number of integer partitions</a>
%H A000041 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.pdf">A combinatorial miscellany</a>
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%H A000041 Yi Wang and Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1303.5595">Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences</a>, arXiv preprint arXiv:1303.5595 [math.CO], 2013.
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%H A000041 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Partition.html">Partition</a>, <a href="https://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>, and <a href="https://mathworld.wolfram.com/RamanujansIdentity.html">Ramanujan's Identity</a>
%H A000041 West Sussex Grid for Learning, Multicultural Mathematics, <a href="http://wsgfl.westsussex.gov.uk/maths/Ramanujan.htm">Ramanujan's Partition of Numbers</a>
%H A000041 Thomas Wieder, <a href="/A000041/a000041.txt">Comment on A000041</a>
%H A000041 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
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%H A000041 Robert M. Ziff, <a href="http://dx.doi.org/10.1088/0305-4470/28/5/013">On Cardy's formula for the critical crossing probability in 2d percolation</a>, J. Phys. A. 28, 1249-1255 (1995).
%H A000041 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000041 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A000041 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%H A000041 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A000041 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000041 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.
%F A000041 G.f.: 1 + Sum_{n>=1} x^n/(Product_{k>=n} 1-x^k). - _Joerg Arndt_, Jan 29 2011
%F A000041 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!
%F A000041 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).
%F A000041 a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy and Ramanujan). See A050811.
%F A000041 a(n) = a(0)*b(n) + a(1)*b(n-2) + a(2)*b(n-4) + ... where b = A000009.
%F A000041 From _Jon E. Schoenfield_, Aug 17 2014: (Start)
%F A000041 It appears that the above approximation from Hardy and Ramanujan can be refined as
%F A000041 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
%F A000041 c0 = -0.230420145062453320665537
%F A000041 c1 = -0.0178416569128570889793
%F A000041 c2 = 0.0051329911273
%F A000041 c3 = -0.0011129404
%F A000041 c4 = 0.0009573,
%F A000041 as n -> infinity. (End)
%F A000041 From _Vaclav Kotesovec_, May 29 2016 (c4 added Nov 07 2016): (Start)
%F A000041 c0 = -0.230420145062453320665536704197233... = -1/36 - 2/Pi^2
%F A000041 c1 = -0.017841656912857088979502135349949... = 1/(6*sqrt(6)*Pi) - sqrt(3/2)/Pi^3
%F A000041 c2 = 0.005132991127342167594576391633559... = 1/(2*Pi^4)
%F A000041 c3 = -0.001112940489559760908236602843497... = 3*sqrt(3/2)/(4*Pi^5) - 5/(16*sqrt(6)*Pi^3)
%F A000041 c4 = 0.000957343284806972958968694349196... = 1/(576*Pi^2) - 1/(24*Pi^4) + 93/(80*Pi^6)
%F A000041 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).
%F A000041 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).
%F A000041 (End)
%F A000041 a(n) < exp( (2/3)^(1/2) Pi sqrt(n) ) (Ayoub, p. 197).
%F A000041 G.f.: Product_{m>=1} (1+x^m)^A001511(m). - _Vladeta Jovovic_, Mar 26 2004
%F A000041 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
%F A000041 G.f.: Product_{i>=1} Product_{j>=0} (1+x^((2i-1)*2^j))^(j+1). - _Jon Perry_, Jun 06 2004
%F A000041 G.f. e^(Sum_{k>0} (x^k/(1-x^k)/k)). - _Franklin T. Adams-Watters_, Feb 08 2006
%F A000041 a(n) = A114099(9*n). - _Reinhard Zumkeller_, Feb 15 2006
%F A000041 Euler transform of all 1's sequence (A000012). Weighout transform of A001511. - _Franklin T. Adams-Watters_, Mar 15 2006
%F A000041 a(n) = A027187(n) + A027193(n) = A000701(n) + A046682(n). - _Reinhard Zumkeller_, Apr 22 2006
%F A000041 A026820(a(n),n) = A134737(n) for n > 0. - _Reinhard Zumkeller_, Nov 07 2007
%F A000041 Convolved with A152537 gives A000079, powers of 2. - _Gary W. Adamson_, Dec 06 2008
%F A000041 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
%F A000041 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
%F A000041 From _Jerome Malenfant_, Feb 14 2011: (Start)
%F A000041 a(n) = determinant of the n X n Toeplitz matrix:
%F A000041 1 -1
%F A000041 1 1 -1
%F A000041 0 1 1 -1
%F A000041 0 0 1 1 -1
%F A000041 -1 0 0 1 1 -1
%F A000041 . . .
%F A000041 d_n d_(n-1) d_(n-2)...1
%F A000041 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)
%F A000041 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 A000041 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
%F A000041 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
%F A000041 a(n) = A035363(2n). - _Omar E. Pol_, Nov 20 2009
%F A000041 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
%F A000041 Convolution of A010815 with A000712. - _Gary W. Adamson_, Jul 20 2012
%F A000041 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
%F A000041 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
%F A000041 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
%F A000041 G.f.: 1/(x; x)_{inf} where (a; q)_k is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Apr 24 2013
%F A000041 a(n) = A066186(n)/n, n >= 1. - _Omar E. Pol_, Aug 16 2013
%F A000041 From _Peter Bala_, Dec 23 2013: (Start)
%F A000041 a(n-1) = Sum_{parts k in all partitions of n} mu(k), where mu(k) is the arithmetical Möbius function (see A008683).
%F A000041 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).
%F A000041 n*( a(n) - a(n-1) ) = Sum_{parts k in all partitions in P(2,n)} k (see A138880).
%F A000041 Let P(3,n) denote the set of partitions of n into parts k >= 3. Then
%F A000041 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)
%F A000041 a(n) = Sum_{k=1..n+1} (-1)^(n+1-k)*A000203(k)*A002040(n+1-k). - _Mircea Merca_, Feb 27 2014
%F A000041 a(n) = A240690(n) + A240690(n+1), n >= 1. - _Omar E. Pol_, Mar 16 2015
%F A000041 From _Gary W. Adamson_, Jun 22 2015: (Start)
%F A000041 A production matrix for the sequence with offset 1 is M, an infinite n x n matrix of the following form:
%F A000041 a, 1, 0, 0, 0, 0, ...
%F A000041 b, 0, 1, 0, 0, 0, ...
%F A000041 c, 0, 0, 1, 0, 0, ...
%F A000041 d, 0, 0, 0, 1, 0, ...
%F A000041 .
%F A000041 .
%F A000041 ... such that (a, b, c, d, ...) is the signed version of A080995 with offset 1: (1,1,0,0,-1,0,-1,...)
%F A000041 and a(n) is the upper left term of M^n.
%F A000041 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)
%F A000041 G.f.: x^(1/24)/eta(log(x)/(2 Pi i)). - _Thomas Baruchel_, Jan 09 2016, after _Michael Somos_ (after Richard Dedekind).
%F A000041 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
%F A000041 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
%F A000041 a(n) ~ 2*Pi * BesselI(3/2, sqrt(24*n-1)*Pi/6) / (24*n-1)^(3/4). - _Vaclav Kotesovec_, Jan 11 2017
%F A000041 G.f.: Product_{k>=1} (1 + x^k)/(1 - x^(2*k)). - _Ilya Gutkovskiy_, Jan 23 2018
%F A000041 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
%F A000041 Sum_{n>=1} 1/a(n) = A078506. - _Amiram Eldar_, Nov 01 2020
%F A000041 Sum_{n>=0} a(n)/2^n = A065446. - _Amiram Eldar_, Jan 19 2021
%F A000041 From _Simon Plouffe_, Mar 12 2021: (Start)
%F A000041 Sum_{n>=0} a(n)/exp(Pi*n) = 2^(3/8)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/24)).
%F A000041 Sum_{n>=0} a(n)/exp(2*Pi*n) = 2^(1/2)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/12)).
%F A000041 [corrected by _Vaclav Kotesovec_, May 12 2023] (End)
%F A000041 [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]
%F A000041 a(n) = A000009(n) + A035363(n) + A006477(n). - _R. J. Mathar_, Feb 01 2022
%F A000041 a(n) = A008284(2*n,n) is also the number of partitions of 2n into n parts. - _Ryan Brooks_, Jun 11 2022
%F A000041 a(n) = A000700(n) + A330644(n). - _R. J. Mathar_, Jun 15 2022
%F A000041 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
%e A000041 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
%e A000041 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 + ...
%e A000041 G.f. = 1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ...
%e A000041 From _Gregory L. Simay_, Nov 08 2015: (Start)
%e A000041 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>).
%e A000041 The partition 8,8,8,8 is counted in a(32,4,<4>).
%e A000041 The partition 9,9,9,5 is counted in a(32,4,<3,1>).
%e A000041 The partition 11,11,5,5 is counted in a(32,4,<2,2>).
%e A000041 The partition 13,13,5,1 is counted in a(32,4,<2,1,1>).
%e A000041 The partition 14,9,6,3 is counted in a(32,4,<1,1,1,1>).
%e A000041 a(n odd,4,<2,2>) = 0.
%e A000041 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.
%e A000041 (End)
%p A000041 A000041 := n -> combinat:-numbpart(n): [seq(A000041(n), n=0..50)]; # Warning: Maple 10 and 11 give incorrect answers in some cases: A110375.
%p A000041 spec := [B, {B=Set(Set(Z,card>=1))}, unlabeled ];
%p A000041 [seq(combstruct[count](spec, size=n), n=0..50)];
%p A000041 with(combstruct):ZL0:=[S,{S=Set(Cycle(Z,card>0))}, unlabeled]: seq(count(ZL0,size=n),n=0..45); # _Zerinvary Lajos_, Sep 24 2007
%p A000041 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
%p A000041 # Using the function EULER from Transforms (see link at the bottom of the page).
%p A000041 1,op(EULER([seq(1,n=1..49)])); # _Peter Luschny_, Aug 19 2020
%t A000041 Table[ PartitionsP[n], {n, 0, 45}]
%t A000041 a[ n_] := SeriesCoefficient[ q^(1/24) / DedekindEta[ Log[q] / (2 Pi I)], {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t A000041 a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t A000041 CoefficientList[1/QPochhammer[q] + O[q]^100, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%t A000041 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 *)
%o A000041 (Magma) a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];
%o A000041 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x * O(x^n)), n))};
%o A000041 (PARI) /* The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): */
%o A000041 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))
%o A000041 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))))
%o A000041 g(h, q) = if(q<3,0,sum(k=1,q-1,k*(frac(h*k/q)-1/2)))
%o A000041 part(n) = round(sum(q=1,max(5,0.5*sqrt(n)),L(n,q)*Psi(n,q)))
%o A000041 /* _Ralf Stephan_, Nov 30 2002, fixed by _Vaclav Kotesovec_, Apr 09 2018 */
%o A000041 (PARI) {a(n) = numbpart(n)};
%o A000041 (PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), x^k^2 / prod( i=1, k, 1 - x^i, 1 + x * O(x^n))^2, 1), n))};
%o A000041 (PARI) f(n)= my(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]<n, i=2;while(v[i]==0,i++);v[i]--;s=sum(k=i,n,k*v[k]); while(i>1,i--;s+=i*(v[i]=(n-s)\i));t++);t \\ _Thomas Baruchel_, Nov 07 2005
%o A000041 (PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)/k, x*O(x^n))), n)) \\ _Joerg Arndt_, Apr 16 2010
%o A000041 (MuPAD) combinat::partitions::count(i) $i=0..54 // _Zerinvary Lajos_, Apr 16 2007
%o A000041 (Sage) [number_of_partitions(n) for n in range(46)] # _Zerinvary Lajos_, May 24 2009
%o A000041 (Sage)
%o A000041 @CachedFunction
%o A000041 def A000041(n):
%o A000041 if n == 0: return 1
%o A000041 S = 0; J = n-1; k = 2
%o A000041 while 0 <= J:
%o A000041 T = A000041(J)
%o A000041 S = S+T if is_odd(k//2) else S-T
%o A000041 J -= k if is_odd(k) else k//2
%o A000041 k += 1
%o A000041 return S
%o A000041 [A000041(n) for n in range(50)] # _Peter Luschny_, Oct 13 2012
%o A000041 (Sage) # uses[EulerTransform from A166861]
%o A000041 a = BinaryRecurrenceSequence(1, 0)
%o A000041 b = EulerTransform(a)
%o A000041 print([b(n) for n in range(50)]) # _Peter Luschny_, Nov 11 2020
%o A000041 (Haskell)
%o A000041 import Data.MemoCombinators (memo2, integral)
%o A000041 a000041 n = a000041_list !! n
%o A000041 a000041_list = map (p' 1) [0..] where
%o A000041 p' = memo2 integral integral p
%o A000041 p _ 0 = 1
%o A000041 p k m = if m < k then 0 else p' k (m - k) + p' (k + 1) m
%o A000041 -- _Reinhard Zumkeller_, Nov 03 2015, Nov 04 2013
%o A000041 (Maxima) num_partitions(60,list); /* _Emanuele Munarini_, Feb 24 2014 */
%o A000041 (GAP) List([1..10],n->Size(OrbitsDomain(SymmetricGroup(IsPermGroup,n),SymmetricGroup(IsPermGroup,n),\^))); # _Attila Egri-Nagy_, Aug 15 2014
%o A000041 (Perl) use ntheory ":all"; my @p = map { partitions($_) } 0..100; say "[@p]"; # _Dana Jacobsen_, Sep 06 2015
%o A000041 (Racket)
%o A000041 #lang racket
%o A000041 ; SUM(k,-inf,+inf) (-1)^k p(n-k(3k-1)/2)
%o A000041 ; For k outside the range (1-(sqrt(1-24n))/6 to (1+sqrt(1-24n))/6) argument n-k(3k-1)/2 < 0.
%o A000041 ; Therefore the loops below are finite. The hash avoids repeated identical computations.
%o A000041 (define (p n) ; Nr of partitions of n.
%o A000041 (hash-ref h n
%o A000041 (λ ()
%o A000041 (define r
%o A000041 (+
%o A000041 (let loop ((k 1) (n (sub1 n)) (s 0))
%o A000041 (if (< n 0) s
%o A000041 (loop (add1 k) (- n (* 3 k) 1) (if (odd? k) (+ s (p n)) (- s (p n))))))
%o A000041 (let loop ((k -1) (n (- n 2)) (s 0))
%o A000041 (if (< n 0) s
%o A000041 (loop (sub1 k) (+ n (* 3 k) -2) (if (odd? k) (+ s (p n)) (- s (p n))))))))
%o A000041 (hash-set! h n r)
%o A000041 r)))
%o A000041 (define h (make-hash '((0 . 1))))
%o A000041 ; (for ((k (in-range 0 50))) (printf "~s, " (p k))) runs in a moment.
%o A000041 ; _Jos Koot_, Jun 01 2016
%o A000041 (Python)
%o A000041 from sympy.functions.combinatorial.numbers import partition
%o A000041 print([partition(i) for i in range(101)]) # _Joan Ludevid_, May 25 2025
%o A000041 (Julia) # DedekindEta is defined in A000594
%o A000041 A000041List(len) = DedekindEta(len, -1)
%o A000041 A000041List(50) |> println # _Peter Luschny_, Mar 09 2018
%Y A000041 Cf. A000009, A000079, A000203, A001318, A008284, A026820, A065446, A078506, A113685, A132311, A000248, A000110.
%Y A000041 Partial sums give A000070.
%Y A000041 For successive differences see A002865, A053445, A072380, A081094, A081095.
%Y A000041 Antidiagonal sums of triangle A092905. a(n) = A054225(n,0).
%Y A000041 Boustrophedon transforms: A000733, A000751.
%Y A000041 Cf. A167376 (complement), A061260 (multisets), A000700 (self-conjug), A330644 (not self-conj).
%K A000041 core,easy,nonn,nice
%O A000041 0,3
%A A000041 _N. J. A. Sloane_
%E A000041 Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001
%E A000041 Additional comments from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001