A117953 Number of partitions of n into odd parts and such that parts having size k occur at most k times.

1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 9, 10, 12, 14, 16, 19, 21, 24, 28, 32, 36, 41, 47, 53, 59, 67, 76, 85, 96, 108, 121, 135, 151, 169, 188, 210, 235, 261, 289, 322, 357, 395, 438, 485, 536, 592, 654, 721, 795, 876, 963, 1059, 1165, 1279, 1405, 1541, 1688, 1851

Offset

0,7

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.=product((1-x^(2k(2k-1)))/(1-x^(2k-1)), k=1..infinity).

a(n) ~ Pi^(1/4) * exp(Pi*sqrt(n/3) - 3^(1/4)*zeta(3/2)*n^(1/4)/2^(3/2) - 3*zeta(3/2)^2/(64*Pi)) / (2^(5/4) * 3^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 11 2026

Example

a(10)=4 because we have [9,1],[7,3],[5,5] and [3,3,3,1].

Maple

g:=product((1-x^(2*k*(2*k-1)))/(1-x^(2*k-1)), k=1..50): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..67);

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 - x^(2*k*(2*k-1))) / (1 - x^(2*k-1)), {k, 1, Floor[nmax/2]+1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2026 *)

Crossrefs

Sequence in context: A112176 A112205 A369573 * A128331 A385318 A084827

Adjacent sequences: A117950 A117951 A117952 * A117954 A117955 A117956

Keyword

nonn

Author

Emeric Deutsch, Apr 05 2006

Status

approved