A307606 - OEIS

%I #9 Mar 19 2021 12:07:54

%S 1,2,6,16,46,104,268,596,1406,3060,6812,14356,30948,63660,132328,

%T 267164,541678,1072000,2127052,4140340,8060588,15458948,29602504,

%U 55990780,105693252,197422424,367793952,679206200,1250557768,2284986580,4162202864,7530956532,13583095710

%N G.f. A(x) satisfies: A(x) = ((1 + x)/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

%C Convolution of A307604 and A307605.

%H Vaclav Kotesovec, <a href="/A307606/b307606.txt">Table of n, a(n) for n = 0..5000</a>

%F G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*A074206(k)).

%F a(n) ~ ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(50*(2+r))) * exp(12/625 + 2^(1/(2+r) - 1) * (2+r) * ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(2+r)) / (zeta'(r)^(1/(2+r)) * (1+r)) * n^((1+r)/(2+r))) / (A^(144/625) * 2^((3 + 2*r)/(50*(2 + r))) * zeta'(r)^(1/(50*(2+r))) * sqrt(Pi*(2+r)) * n^(1/2 + 1/(50*(2+r)))), where r = A107311 is the root of the equation zeta(r)=2 and A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Mar 18 2021

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 46*x^4 + 104*x^5 + 268*x^6 + 596*x^7 + 1406*x^8 + 3060*x^9 + ...

%t terms = 32; A[_] = 1; Do[A[x_] = (1 + x)/(1 - x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

%Y Cf. A050369, A074206, A307604, A307605, A318767.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 18 2019