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
Keywords
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).