A027190 Number of partitions of n into an odd number of parts, the least being 4; also, a(n+4) = number of partitions of n into an even number of parts, each >=4.

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 12, 13, 17, 19, 25, 28, 35, 40, 50, 57, 70, 80, 98, 112, 135, 155, 186, 213, 253, 291, 345, 395, 465, 533, 625, 716, 835, 956, 1113, 1272, 1474, 1684, 1946, 2220, 2558, 2915, 3351, 3814, 4372, 4971, 5688, 6457, 7370, 8359, 9524, 10787

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A027196.

Programs

Mathematica

nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 - x^(2*k))*(1 - x^(2*k - 1))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += x^(8*k)/p;, {k, 1, nmax}]; Join[{0, 0, 0}, CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 20 2025 *)

Formula

G.f.: x^4 * Sum_{k>=0} x^(8*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3 * 2^(7/2) * n^(5/2)). - Vaclav Kotesovec, Jun 20 2025

Extensions

More terms from Seiichi Manyama, May 15 2023