A361535 - OEIS

%I #16 Mar 19 2023 14:38:34

%S 1,10,61,290,1172,4212,13833,42262,121625,332764,871641,2197936,

%T 5359005,12679730,29200593,65617892,144189054,310400110,655669910,

%U 1360910666,2779007594,5589070978,11081585154,21679798590,41883282555,79958881544,150943109191,281926365224

%N Expansion of g.f. 1 / Product_{n>=1} ((1 - x^n)^6 * (1 - x^(2*n-1))^4).

%H Vaclav Kotesovec, <a href="/A361535/b361535.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ exp(4*Pi*sqrt(n/3)) / (2^(5/2) * 3^(7/4) * n^(9/4)). - _Vaclav Kotesovec_, Mar 19 2023

%e G.f.: A(x) = 1 + 10*x + 61*x^2 + 290*x^3 + 1172*x^4 + 4212*x^5 + 13833*x^6 + 42262*x^7 + 121625*x^8 + 332764*x^9 + 871641*x^10 + ...

%e A related series begins

%e A(x)^(1/2) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + ... + A360191(n)*x^n + ...

%t nmax = 30; CoefficientList[Series[Product[1/((1 - x^k)^6 * (1 - x^(2*k-1))^4), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 19 2023 *)

%o (PARI) {a(n) = polcoeff( 1/prod(m=1,n, (1 - x^m)^6 * (1 - x^(2*m-1))^4 + x*O(x^n)), n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A360191, A361050, A361550.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 18 2023