A027191 Number of partitions of n into an odd number of parts, the least being 5; also, a(n+5) = number of partitions of n into an even number of parts, each >=5.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 13, 14, 18, 20, 25, 28, 35, 39, 48, 54, 65, 74, 89, 100, 119, 135, 160, 181, 213, 241, 282, 320, 372, 422, 490, 554, 641, 726, 836, 946, 1087, 1229, 1408, 1591, 1817, 2052, 2341, 2639, 3002, 3384

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A026798, A027197.

Programs

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];
a027197[n_] := If[n < 5, 0, b[n - 5, 5, 0]];
a026798[n_] := SeriesCoefficient[x^5/QPochhammer[x^5, x], {x, 0, n}];
a[n_] := a026798[n] - a027197[n];
a /@ Range[55] (* Jean-François Alcover, Feb 06 2020, after Alois P. Heinz in A027197 *)

Formula

a(n) = A026798(n) - A027197(n). - Jean-François Alcover, Feb 06 2020

G.f.: x^5 * Sum_{k>=0} x^(10*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023

a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (4*3^(3/2)*n^3). - Vaclav Kotesovec, Jun 20 2025

Extensions

More terms from Seiichi Manyama, May 15 2023