A370735 - OEIS

%I #8 Feb 29 2024 06:23:46

%S 1,15,1050,52125,3277500,179801250,11966690625,738318187500,

%T 49788716718750,3314446448437500,227432073022265625,

%U 15631633385109375000,1090877899335878906250,76338563689129101562500,5384934139819611328125000,381204340327212964599609375,27111589537137988341064453125

%N a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

%C In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} 1/(1 - d*x^k)^(1/m), then a(n) ~ d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

%F G.f.: Product_{k>=1} 1/(1 - 3*(25*x)^k)^(1/5).

%F a(n) ~ 75^n / (Gamma(1/5) * QPochhammer(1/3)^(1/5) * n^(4/5)).

%t nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]

%t nmax = 20; CoefficientList[Series[Product[1/(1-3*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

%Y Cf. A242587 (d=3,m=1), A370714 (d=3,m=2), A370710 (d=3,m=3), A370734 (d=3,m=4).

%Y Cf. A070933 (d=2,m=1), A370713 (d=2,m=2), A370715 (d=2,m=3), A370732 (d=2,m=4), A370733 (d=2,m=5).

%Y Cf. A000041 (d=1,m=1), A271235 (d=1,m=2), A271236 (d=1,m=3).

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Feb 28 2024