A377108 - OEIS

%I #7 Oct 18 2024 03:53:58

%S 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,

%T 9,11,12,15,16,20,22,26,29,35,37,43,47,55,60,70,77,90,100,115,128,149,

%U 165,190,212,242,269,306,339,385,427,482,536,606,672,757

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

%C 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)).

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

%H Vaclav Kotesovec, <a href="/A377108/b377108.txt">Table of n, a(n) for n = 0..10000</a>

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

%t 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]

%t 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]]

%Y Column 7 of A350879.

%Y Cf. A000041.

%K nonn

%O 0,24

%A _Vaclav Kotesovec_, Oct 16 2024