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%I #11 Mar 19 2021 11:52:49
%S 1,1,3,6,17,28,72,122,282,493,1027,1790,3673,6300,12205,21117,39782,
%T 67989,124937,212189,381705,644625,1136315,1905352,3312916,5513005,
%U 9443362,15624026,26445046,43451200,72751824,118792691,196966722,319714816,525316191,847734183,1381904765
%N G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...
%C Euler transform of A050369.
%H Vaclav Kotesovec, <a href="/A307604/b307604.txt">Table of n, a(n) for n = 0..5000</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*A074206(k)).
%F a(n) ~ (-Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(50*(2+r))) * exp(12/625 + ((2+r)/(1+r)) * (-Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (A^(144/625) * sqrt(2*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 + x + 3*x^2 + 6*x^3 + 17*x^4 + 28*x^5 + 72*x^6 + 122*x^7 + 282*x^8 + 493*x^9 + ...
%t terms = 36; A[_] = 1; Do[A[x_] = 1/(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, A129374, A307605, A307606, A307607.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 18 2019
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