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A000041
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a(n) = number of partitions of n (the partition numbers).
(Formerly M0663 N0244)
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1418
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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;
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listen;
history;
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internal format)
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OFFSET
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0,3
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COMMENTS
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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
This is also the number of parts equal to 1 in the outer shell of the partitions of n+1 (see A138151). - Omar E. Pol, Apr 17 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
a(n) is also the number of partitions of 2n into even parts. More generally, it appears that a(n) is also the number of partitions of k*n into parts divisible by k, for k>0. [From Omar E. Pol, Nov 20 2009, Nov 25 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
a(n) = A027293(n+k,k), the number of partitions of n+k that contain k as a part [Omar E. Pol, Nov 27 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 ordered degree sequences of all trees on vertex set {1,2,...,n+2}. Take a partition of the integer n, add 1 to each part and append 1's so that the total is 2n+2. Now you 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
a(n) is also the sum of Dyson's ranks of all partitions of n+1 that do not contain 1 as a part, if n >= 1. (Cf. A195822). - Omar E. Pol, Nov 06 2011
a(n) is also the number of regions of n (Cf. A186114, A193870). a(n) is also the number of parts in the last region of n (Cf. A194446). It appears that a(n) is also the number of regions of n+2 that contain only one part (Cf. A194439). - Omar E. Pol, Dec 01 2011
It appears that a(n) is also the total number of parts of size k in all outer shells of the partitions of the next k integers (Cf. A135010, A194812, A182703, A182712-A182714). - Omar E. Pol, Feb 03 2012
It appears that a(n) is also the difference, between n+k and n, of the total number of parts of size k in all their partitions, if k >= 1 (Cf. A066633). - Omar E. Pol, Feb 04 2012
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)
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REFERENCES
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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 II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997
B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.
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)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)147-163).
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.
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. [From Parthasarathy Nambi, Sep 21 2008]
Robert M. Young, "Excursions in Calculus", Mathematical Association of America, p. 367. [From Gary W. Adamson, Oct 05 2008]
Robert M. Ziff, "On Cardy's formula for the critical crossing probability in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).
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LINKS
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David W. Wilson, Table of n, a(n) for n = 0..10000
Joerg Arndt, 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 [Broken link corrected by Steve Vonn (5463math(AT)gmail.com), Jan 03 2009]
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
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.
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)
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
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
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. 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
O. E. Pol, How to build a shell model of partitions; A shell model of partitions (2D and 3D); Illustration of initial terms (2D view); Illustration of initial terms (3D view) [From Omar E. Pol, Sep 06 2008]
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
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
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FORMULA
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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).
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 by 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
It appears that a(n) = Sum_{j=1..k} T(n+j,k), where T(n,k) is the number of parts of size k in the outer shell of the partitions of n (Cf. A135010, A182703, A182712-A182714). - Omar E. Pol, Feb 03 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
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EXAMPLE
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1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + ...
1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ...
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MAPLE
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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 (zerinvarylajos(AT)yahoo.com), 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 (zerinvarylajos(AT)yahoo.com), Dec 16 2007
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MATHEMATICA
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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 *)
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PROG
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(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 (zerinvarylajos(AT)yahoo.com), Apr 16 2007
(Sage) [number_of_partitions(n) for n in xrange(0, 46)] # [Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
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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).
Sequence in context: A046054 A092885 A213598 * A218027 A084251 A218026
Adjacent sequences: A000038 A000039 A000040 * A000042 A000043 A000044
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KEYWORD
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core,easy,nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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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.
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STATUS
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approved
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