A058696 Number of ways to partition 2n into positive integers.

1, 2, 5, 11, 22, 42, 77, 135, 231, 385, 627, 1002, 1575, 2436, 3718, 5604, 8349, 12310, 17977, 26015, 37338, 53174, 75175, 105558, 147273, 204226, 281589, 386155, 526823, 715220, 966467, 1300156, 1741630, 2323520, 3087735, 4087968, 5392783, 7089500, 9289091

Offset: 0

Author

N. J. A. Sloane, Dec 31 2000

Keywords

• nonn

Comments

A bisection of A000041, the other one is A058695.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Feb 16 2014

a(n) is the number of partitions of 3n-2 having n as a part, for n >= 1. Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1. - Clark Kimberling, Mar 02 2014

Examples

			G.f. = 1 + 2*x + 5*x^2 + 11*x^3 + 22*x^4 + 42*x^5 + 77*x^6 + 135*x^7 + ...
G.f. = q^-1 + 2*q^47 + 5*q^95 + 11*q^143 + 22*q^191 + 42*q^239 + 77*q^287 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
• K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
• Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000041, A035294, A058695.

Programs

Maple

a:= n-> combinat[numbpart](2*n):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 29 2020

Mathematica

nn=100;Table[CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x][[2i-1]],{i,1,nn/2}] (* Geoffrey Critzer, Sep 28 2013 *)
(* also *)
Table[PartitionsP[2 n], {n, 0, 40}] (* Clark Kimberling, Mar 02 2014 *)
(* also *)
Table[Count[IntegerPartitions[3 n - 2], p_ /; MemberQ[p, n]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(8*k-4))*(1 + x^(8*k))*(1 + x^k)^2/((1 + x^(8*k-1))*(1 + x^(8*k-7))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2016 *)

MuPAD

combinat::partitions::count(2*i) $i=0..54 // Zerinvary Lajos, Apr 16 2007

PARI

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 1))), 2*n))}; /* Michael Somos, Apr 25 2003 */

PARI

a(n) = numbpart(2*n); \\ Michel Marcus, Sep 28 2013

Formula

Expansion of f(x^3, x^5) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 16 2014

Euler transform of period 16 sequence [ 2, 2, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, ...]. - Michael Somos, Apr 25 2003

a(n) = A000041(2*n).

Convolution of A000041 and A035294. - Michael Somos, Feb 16 2014

G.f.: Product_{k>=1} (1 + x^(8*k-4)) * (1 + x^(8*k)) * (1 + x^k)^2 / ((1 + x^(8*k-1)) * (1 + x^(8*k-7)) * (1 - x^k)). - Vaclav Kotesovec, Nov 17 2016

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Feb 16 2022