A027192 Number of partitions of n into an odd number of parts, the least being 6; also, a(n+6) = number of partitions of n into an even number of parts, each >=6.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 14, 15, 19, 21, 26, 29, 35, 39, 48, 53, 63, 71, 84, 94, 111, 124, 145, 163, 189, 212, 247, 276, 318, 358, 411, 461, 529, 593, 678, 761, 866, 971, 1106, 1238, 1404, 1574, 1781, 1993

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

In general, if m>=1 and g.f. = x^m * Sum_{k>=0} x^(2*m*k) / Product_{j=1..2*k} (1-x^j), then a(n) ~ Pi^(m-1) * (m-1)! * exp(Pi*sqrt(2*n/3)) / (2^((m+5)/2) * 3^(m/2) * n^((m+1)/2)). - Vaclav Kotesovec, Jun 20 2025

Links

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

Crossrefs

Cf. A027187, A027188, A027189, A027190, A027191, A027198.

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^(12*k)/p;, {k, 1, nmax}]; Join[{0, 0, 0, 0, 0}, CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 20 2025 *)

Formula

G.f.: x^6 * Sum_{k>=0} x^(12*k)/Product_{j=1..2*k} (1-x^j). - Seiichi Manyama, May 15 2023

a(n) ~ 5 * Pi^5 * exp(Pi*sqrt(2*n/3)) / (9 * 2^(5/2) * n^(7/2)). - Vaclav Kotesovec, Jun 20 2025

Extensions

More terms from Vladeta Jovovic, Aug 01 2009