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A000041 a(n) = number of partitions of n (the partition numbers).
(Formerly M0663 N0244)
1840
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also number of nonnegative solutions to b+2c+3d+4e+...=n and the number of nonnegative solutions to 2c+3d+4e+...<=n. - Henry Bottomley, Apr 17 2001

a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).

Also the number of rooted trees with n+1 nodes and height at most 2.

Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras gl(n). A006950, A015128 and this sequence together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003

a(n)=a(0)b(n)+a(1)b(n-2)+a(2)b(n-4)+... where b=A000009.

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

It is unknown if there are infinitely many partition numbers divisible by 3, although it is known that there are infinitely many divisible by 2. - Jonathan Vos Post, Jun 21 2005

Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 07 2005

a(n) = A114099(9*n). - Reinhard Zumkeller, Feb 15 2006

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..inf} b_j z^j, b_0 != 0. Then 1/P(z) = Sum{j=0..inf} 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

A026820(a(n),n) = A134737(n) for n>0. - Reinhard Zumkeller, Nov 07 2007

Equals row sums of triangle A137683. - Gary W. Adamson, Feb 05 2008

a(n)= the number of different ways to run up a staircase with n steps, taking steps of sizes 1,2,3,... and r (r<=n), where the order is not important and there is no restriction on the number or the size of each step taken. - Mohammad K. Azarian, May 21 2008

Equals the eigenvector of triangle A145006 and row sums of the eigentriangle of the partition numbers, A145007. - Gary W. Adamson, Sep 28 2008

Starting with offset 1 = INVERT transform of (1, 1, 0, 0, -1, 0, -1,...), where A080995, the characteristic function of A001318 (1, 2, 5, 7, 12,...) is signed (++ -- ++,...) as to 1's. This is equivalent to Lim__{n=1..inf} A145006^n as a vector. The INVERT transform of (1, 1, 0, 0, -1,...) begins (1, 2,..) then for each successive operation we take a dot product of (1, 1, 0, 0, -1,...) in reverse and the ongoing results of our series (1, 2, 3, 5, 7,...) then add the result to the next term in (1, 1, 0, 0, -1,...). For example, a (7) = 15 = (0, -1, 0, 0, 1, 1) dot (1, 2, 3, 5, 7, 11) = (0*1, (-1)*2, 0*3, 0*5, 1*7, 1*11) = (-2 + 7 + 11) = 16, then add to (-1) = 15. - Gary W. Adamson, Oct 05 2008

Convolved with A147843 = A000203 prefaced with a zero: (0, 1, 3, 4, 7,...). - Gary W. Adamson, Nov 15 2008

Equals an infinite convolution product_(1,1,1,...)*(1,0,1,0,1,...)* (1,0,0,1,0,0,1,...)*(1,0,0,0,1,0,0,0,1,...)* ...; = a*b*c*...; where a = (1/(1-x)), b = (1/(1-x^2)), c = (1/(1-x^3)), ...etc. An array by rows: row 1 = a, row 2 = a*b, row 3 = a*b*c,...; gives:

  1, 1, 1, 1, 1, 1,. 1,. 1,. 1,..1,... = (a).................................

  1, 1, 2, 2, 3, 3,. 4,..4,. 5,..5,... = (a*b)...............................

  1, 1, 2, 3, 4, 5,. 7,..8,.10,.11,... = (a*b*c).............................

  1, 1, 2, 3, 4, 5,. 6,..9,.11,.17,... = (a*b*c*d)...........................

  1, 1, 2, 3, 5, 5,. 7,.10,.13,.18,... = (a*b*c*d*e).........................

  1, 1, 2, 3, 5, 7,.11,.14,.20,.25,... = (a*b*c*d*e*f).......................

  1, 1, 2, 3, 5, 7,.11,.15,.21,.27,... = (a*b*c*d*e*f*g).....................

  1, 1, 2, 3, 5, 7,.11,.15,.22,.28,... = (a*b*c*d*e*f*g*h)...................

  1, 1, 2, 3, 5, 7,.11,.15,.22,.29,... = (a*b*c*d*e*f*g*h*i).................

  ... with rows tending to A000041. Partition triangles A058398 = ascending antidiagonals. Partition triangle A008284 reversal of A058398. - Gary W. Adamson, Jun 12 2009

Starting with offset 1 = row sums of triangle A168532. - Gary W. Adamson, Nov 28 2009

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

P(x) = A(x)/A(x^2) with P(x) = (1+x+2x^2+3x^3+5x^4+7x^5 + ...),

  and A(x) = (1+x+3x^2+4x^3+10x^4+13x^5 + ...),

  and A(x^2) = (1+x^2+3x^4+4x^6+10x^8+ ...), where A092119 = (1, 1, 3, 4, 10,...) = Euler transform of the ruler sequence, A001511. - Gary W. Adamson, Feb 11 2010

Equals row sums of triangle A173304. - Gary W. Adamson, Feb 15 2010

p(x) = A(x)*A(x^2), A(x) = A174065; p(x) = B(x)*B(x^3), B(x) = A174068. Equals row sums of triangles A174066 and A174067. - Gary W. Adamson, Mar 06 2010

Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010

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

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. - Jerome Malenfant, Feb 14 2011

Equals row sums of triangle A187566. - Gary W. Adamson, Mar 21 2011

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

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

G. E. Andrews & 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.

B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.

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.

Lynn Chua and Krishanu Roy Sankar, Equipopularity Classes of 132-Avoiding Permutations, The Electronic Journal of Combinatorics 21(1)(2014), #P59, http://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i1p59/pdf. [Cited by Shalosh B. Ekhad and Doron Zeilberger, 2014] - N. J. A. Sloane, Mar 31 2014

L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp. 101-164,Chelsea NY 1992.

F. J. Dyson, Some guesses in the theory of partitions, Eureka, 8 (1944), 10-15.

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 formulae in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth  edition), Oxford Univ. Press (Clarendon), 1979, 273-296.

J. M. Kane, Distribution of orders of Abelian groups, Math. Mag., 49 (1976), 132-135.

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. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.

J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.

Kate Rudolph, Pattern Popularity in 132-Avoiding Permutations, The Electronic Journal of Combinatorics 20(1)(2013), #P8, http://www.combinatorics.org/ojs/index.php/eljc/article/download/v20i1p8/pdf. [Cited by Shalosh B. Ekhad and Doron Zeilberger, 2014] - N. J. A. Sloane, Mar 31 2014

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.

Robert M. Young, "Excursions in Calculus", Mathematical Association of America, p. 367.

Robert M. Ziff, "On Cardy's formula for the critical crossing probability in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).

LINKS

David W. Wilson, Table of n, a(n) for n = 0..10000

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4, pp.344-353

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

S. Ahlgren and K. Ono, Addition and Counting: The Arithmetic of Partitions

S. Ahlgren & K. Ono, Congruence properties for the partition function

S. Ahlgren & K. Ono, Congruence properties for the partition function

G. Almkvist, Asymptotic Formulas and Generalized Dedekind Sums

Almkvist, Gert, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.

G. Almkvist and H. S. Wilf, On the coefficients in the Hardy-Ramanujan-Rademacher formula for p(n)

Amazing Mathematical Object Factory, Information on Partitions

G. E. Andrews, Three Aspects of Partitions

G. E. Andrews, On a Partition Function of Richard Stanley.

G. E. Andrews & K. Ono, Ramanujan's congruences and Dyson's crank

G. E. Andrews & R. Roy, Ramanujan's Method in q-series Congruences

Anonymous, Bibliography on Partitions

A. O. L. Atkins & F. G. Garvan, Relations between the ranks and cranks of partitions

A. Berkovich & F. G. Garvan, On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5

A. Berkovich & F. G. Garvan, On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations

B. C. Berndt, Ramanujan's congruences for the partition function modulo 5,7 and 11

B. C. Berndt and K. Ono, Ramanujan's Unpublished Manuscript On The Partition And Tau Functions With Proofs And Commentary

B. C. Berndt and K. Ono, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

H. Bottomley, Illustration of initial terms

H. Bottomley, Illustration of initial terms of A000009, A000041 and A047967

H. Bottomley, Partition and composition calculator

K. S. Brown, Additive Partitions of Numbers

K. 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

J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

P. 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.

J. Davis & E. Perez, Computations Of The Partition Function, p(n)

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, 2014.

N. J. Fine, Some New Results On Partitions

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 41

A. Folsom, Z. A. Kent and K. Ono, ''l''-adic properties of the partition function. In press.

A. Folsom, Z. A. Kent and K. Ono, ''l''-adic properties of the partition function, Adv. Math. 229 (2012) 1586

B. Forslund, Partitioning Integers

H. Fripertinger, Partitions of an Integer

GEO magazine, Zahlenspalterei

A. Hassen and T. J. Olsen, Playing With Partitions On The Computer

A. D. Healy, Partition Identities

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 61

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 74

Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, (2009), arXiv:0909.2331v1.

E. 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.

J. Laurendi, Partitions of Integers

Oleg Lazarev, Matt Mizuhara, Ben Reid, Some Results in Partitions, Plane Partitions, and Multipartitions

T. Lockette, Explore Magazine, Path To Partitions

J. Malenfant, Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers

Dr. Math, Partitioning the Integers

Dr. Math, 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

Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.

J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187.

G. P. Michon, Table of partition function p(n) (n=0 through 4096)

G. P. Michon, Partition function

G. A. Miller, Number of the abelian groups of a given order

Hisanori Mishima, Factorization of Partition Numbers

D. J. Newman, A simplified proof of the partition formula, Michigan Math. J. 9:3 (1962), pp. 193-287.

K. Ono, Arithmetic of the partition function

K. Ono, Parity of the partition function

K. Ono, Distribution of the partition function modulo m, Annals Math. 151 (2000), 293-307

K. 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

I. Peterson, The Power Of Partitions

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

M. Planat, Quantum 1/f Noise in Equilibrium: from Planck to Ramanujan

Simon Plouffe, Partitions [Contains first 10000000 terms]

Simon Plouffe, Partition numbers through n = 300000; Partitions numbers from 300000 to 450000; Partitions numbers from 450000 to 500000

M. Presern, Some Results On Partitions

W. A. Pribitkin, The Ramanujan Journal 4(4) 2000, Revisiting Rademacher's Formula for the Partition Function p(n)

PYTHAGORAS, Ramanujan and The Partition Function(Text in Dutch)

S. Ramanujan, Some Properties Of p(n), The Number Of Partitions Of n

S. Ramanujan, Congruence Properties Of Partitions

S. Ramanujan, Congruence Properties Of Partitions

S. Ramanujan & G. H. Hardy, Une formule asymptotique pour le nombre de partitions de n

J. D. Rosenhouse, Partitions of Integers

J. D. Rosenhouse, Solutions to Problems

F. Ruskey, Generate Numerical Partitions

F. Ruskey, The first 284547 partition numbers (52MB compressed file)

M. Savic, The Partition Function and Ramanujan's 5k+4 Congruence

T. Sillke, Number of integer partitions

R. P. Stanley, A combinatorial miscellany

Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595, 2013

R. L. Weaver, The Ramanujan Journal 5(1) 2001, New Congruences for the Partition Function

Eric Weisstein's World of Mathematics, Partition

Eric Weisstein's World of Mathematics, Partition Function P

Eric Weisstein's World of Mathematics, Ramanujan's Identity

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

West Sussex Grid for Learning, Multicultural Mathematics, Ramanujan's Partition of Numbers

Thomas Wieder, Comment on A000041

Wikipedia, Integer Partition

H. S. Wilf, Lectures on Integer Partitions

Wolfram Research, Generating functions of p(n)

D. J. Wright, Partitions

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

FORMULA

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

G.f.: 1+sum(n>=1, x^n/prod(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, 1, ..., 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).

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)

a(n) < exp( (2/3)^(1/2) Pi sqrt(n) ) (Ayoub, p. 197).

G.f.: Product (1+x^m)^A001511(m); m=1..inf. - 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.: prod(i>=1, prod(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

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

Convolved with A152537 gives A000079, powers of 2. - Gary W. Adamson, Dec 06 2008

a(n) = Tr(n)/(24*n-1) = A183011(n)/A183010(n), n>=1. See the Bruinier-Ono paper in the link. - 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(p(n)*exp(-Pi*x*(n+1)),n=0..infinity), 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 Mobius 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 Merten'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) = 124,754, a(48) = 147,273 and a(49) = 173,525 into the recurrence gives the approximation a(50) ~ 204,252.48... compared with the true value a(50) = 204,226. (End)

a(n)=sum_{k=1..n+1} (-1)^(n+1-k)*A000203(k)*A002040(n+1-k). - Mircea Merca, Feb 27 2014

EXAMPLE

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 + ...

a(5) = 7: {[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

MAPLE

with(combinat); A000041 := numbpart; [ seq(numbpart(i), i=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

MATHEMATICA

Table[ PartitionsP[n], {n, 0, 45}]

a[ n_] := SeriesCoefficient[ q^(1/24) / DedekindEta[ Log[q] / (2 Pi I)], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)

PROG

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

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

(PARI) /* The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): */

Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))

L(n, q) = if(q==1, 1, sum(h=1, q-1, if(gcd(h, q)>1, 0, cos((g(h, q)-2*h*n)*Pi/q))))

g(h, q) = if(q<3, 0, sum(k=1, q-1, k*(frac(h*k/q)-1/2)))

part(n) = round(sum(q=1, max(5, 0.24*sqrt(n)+2), L(n, q)*Psi(n, q)))

/* Ralf Stephan, Nov 30 2002 */

(PARI) {a(n) = numbpart(n)};

(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), x^k^2 / prod( i=1, k, 1 - x^i, 1 + x * O(x^n))^2, 1), n))};

(PARI) f(n)= {local(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 (baruchel(AT)users(AT)sourceforge.net), Nov 07 2005 */

(PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

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

(Sage) [number_of_partitions(n) for n in xrange(0, 46)] # Zerinvary Lajos, May 24 2009

(Sage)

@CachedFunction

def A000041(n):

    if n == 0: return 1

    S = 0; J = n-1; k = 2

    while 0 <= J:

        T = A000041(J)

        S = S+T if is_odd(k//2) else S-T

        J -= k if is_odd(k) else k//2

        k += 1

    return S

[A000041(n) for n in (0..49)]  # Peter Luschny, Oct 13 2012

(Haskell)

a000041 = p 1 where

   p _ 0 = 1

   p k m = if m < k then 0 else p k (m - k) + p (k + 1) m

-- Reinhard Zumkeller, Nov 04 2013

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

(GAP) List([1..10], n->Size(OrbitsDomain(SymmetricGroup(IsPermGroup, n), SymmetricGroup(IsPermGroup, n), \^))); # Attila Egri-Nagy, 2014.08.15.

CROSSREFS

Cf. A000009, A008284, A000203, A001318, A132311, A145006, A145007, A147843, A152537, A000079, A173238, A173239, A173241, A173304, A174065, A174066, A174068, A113685, A176202, A168532.

For successive differences see A002865, A053445, A072380, A081094, A081095.

Antidiagonal sums of triangle A092905. a(n) = A054225(n,0).

Boustrophedon transforms: A000733, A000751.

Sequence in context: A242701 A092885 A213598 * A218027 A241729 A084251

Adjacent sequences:  A000038 A000039 A000040 * A000042 A000043 A000044

KEYWORD

core,easy,nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001 and from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

STATUS

approved

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Last modified October 31 05:13 EDT 2014. Contains 248845 sequences.