A363076 - OEIS

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

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OFFSET

1,8

LINKS

Table of n, a(n) for n=1..61.

FORMULA

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

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 *)

PROG

(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))))

CROSSREFS

Cf. A049820, A237828, A363075, A363077.

Cf. A237826.

Sequence in context: A211542 A368052 A022955 * A254297 A306972 A087279

Adjacent sequences: A363073 A363074 A363075 * A363077 A363078 A363079

KEYWORD

nonn

AUTHOR

Seiichi Manyama, May 17 2023

STATUS

approved