A363077 Number of partitions of n such that 5*(least part) + 1 = greatest part.

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 14, 21, 27, 37, 46, 63, 75, 97, 119, 149, 178, 222, 260, 317, 373, 447, 520, 620, 713, 839, 965, 1123, 1282, 1488, 1687, 1939, 2196, 2508, 2826, 3220, 3610, 4087, 4578, 5157, 5755, 6472, 7199, 8060, 8953, 9991, 11069, 12330, 13625, 15134, 16708, 18508

Offset

1,9

Links

Table of n, a(n) for n=1..59.

Formula

G.f.: Sum_{k>=1} x^(6*k+1)/Product_{j=k..5*k+1} (1-x^j).

Mathematica

nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(5*k - 4))*(1 - x^(5*k - 3))*(1 - x^(5*k - 2))*(1 - x^(5*k - 1))*(1 - x^(5*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(6*k + 1)/(1 - x^k)/(1 - x^(5*k + 1))/p; , {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 19 2025 *)

Prog

(PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k+1)/prod(j=k, 5*k+1, 1-x^j))))

Crossrefs

Cf. A049820, A237828, A363075, A363076.

Cf. A237827.

Sequence in context: A187774 A031216 A004683 * A240305 A100036 A179781

Adjacent sequences: A363074 A363075 A363076 * A363078 A363079 A363080

Keyword

nonn

Author

Seiichi Manyama, May 17 2023

Status

approved