A363075 Number of partitions of n such that 3*(least part) + 1 = greatest part.

0, 0, 0, 0, 1, 1, 2, 3, 6, 6, 10, 12, 18, 20, 27, 32, 42, 47, 59, 67, 85, 94, 113, 126, 152, 169, 198, 220, 257, 282, 326, 359, 413, 452, 512, 563, 639, 695, 781, 853, 958, 1041, 1161, 1261, 1402, 1524, 1685, 1827, 2021, 2186, 2407, 2604, 2861, 3088, 3385, 3657, 4002, 4316, 4704, 5069, 5531

Offset

1,7

Links

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

Formula

G.f.: Sum_{k>=1} x^(4*k+1)/Product_{j=k..3*k+1} (1-x^j).

a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 0.33761... - Vaclav Kotesovec, Jun 20 2025

Mathematica

nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(3*k - 2))*(1 - x^(3*k - 1))*(1 - x^(3*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(4*k + 1)/(1 - x^k)/(1 - x^(3*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], Vec(sum(k=1, N, x^(4*k+1)/prod(j=k, 3*k+1, 1-x^j))))

Crossrefs

Cf. A049820, A237828, A363076, A363077.

Cf. A237825.

Sequence in context: A328565 A308762 A155215 * A119319 A069808 A131259

Adjacent sequences: A363072 A363073 A363074 * A363076 A363077 A363078

Keyword

nonn

Author

Seiichi Manyama, May 17 2023

Status

approved