A377108 G.f.: Sum_{k>=1} x^(8*k-1) * Product_{j=1..k-1} (1-x^(7*k+j-1))/(1-x^j).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 6, 6, 8, 9, 11, 12, 15, 16, 20, 22, 26, 29, 35, 37, 43, 47, 55, 60, 70, 77, 90, 100, 115, 128, 149, 165, 190, 212, 242, 269, 306, 339, 385, 427, 482, 536, 606, 672, 757

Offset: 0

Author

Vaclav Kotesovec, Oct 16 2024

Keywords

• nonn

Comments

In general, if m > 1 and g.f. = Sum_{k>=1} x^((m+1)*k-1) * Product_{j=1..k-1} (1-x^(m*k+j-1))/(1-x^j), then a(n) ~ m! * Pi^m * exp(Pi*sqrt(2*n/3)) / (2^((m+4)/2) * 3^((m+1)/2) * n^((m+2)/2)).

Equivalently, a(n) ~ m! * Pi^m * A000041(n) / (6^(m/2) * n^(m/2)).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Column 7 of A350879.

Cf. A000041.

Programs

Mathematica

nmax=100; CoefficientList[Series[Sum[x^(8*k-1)*Product[(1-x^(7*k+j-1))/(1-x^j), {j, 1, k-1}], {k, 1, nmax/8+1}], {x, 0, nmax}], x]
nmax=100; p=x^6; s=x^6; Do[p=Normal[Series[p*x^8*(1-x^(8*k-1))*(1-x^(8*k))*(1-x^(8*k+1))*(1-x^(8*k+2))*(1-x^(8*k+3))*(1-x^(8*k+4))*(1-x^(8*k+5))*(1-x^(8*k+6))/((1-x^(7*k+6))*(1-x^(7*k+5))*(1-x^(7*k+4))*(1-x^(7*k+3))*(1-x^(7*k+2))*(1-x^(7*k+1))*(1-x^(7*k))*(1-x^k)), {x, 0, nmax}]]; s+=p;, {k, 1, nmax/8+1}]; Join[{0}, Take[CoefficientList[s, x], nmax]]

Formula

a(n) ~ 35 * Pi^7 * exp(Pi*sqrt(2*n/3)) / (9 * 2^(3/2) * n^(9/2)).