A363076 Number of partitions of n such that 4*(least part) + 1 = greatest part.

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 10, 14, 19, 25, 33, 41, 51, 65, 79, 97, 116, 140, 165, 198, 233, 272, 316, 369, 422, 493, 561, 643, 731, 835, 943, 1072, 1205, 1359, 1524, 1717, 1911, 2147, 2387, 2665, 2960, 3295, 3640, 4049, 4469, 4950, 5455, 6028, 6622, 7310, 8024, 8826, 9676, 10632, 11627, 12765

Offset: 1

Author

Seiichi Manyama, May 17 2023

Keywords

• nonn

Crossrefs

Cf. A049820, A237828, A363075, A363077.

Cf. A237826.

Programs

Mathematica

nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(4*k - 3))*(1 - x^(4*k - 2))*(1 - x^(4*k - 1))*(1 - x^(4*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(5*k + 1)/(1 - x^k)/(1 - x^(4*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 19 2025 *)

PARI

my(N=70, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k+1)/prod(j=k, 4*k+1, 1-x^j))))

Formula

G.f.: Sum_{k>=1} x^(5*k+1)/Product_{j=k..4*k+1} (1-x^j).