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G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...
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%I #9 Mar 19 2021 11:53:39

%S 1,1,2,5,12,20,48,81,169,305,580,1009,1966,3338,6067,10503,18730,

%T 31633,55641,93151,160389,267585,452762,747016,1253644,2049943,

%U 3390786,5516227,9034745,14572790,23668066,37918484,61042425,97231826,155292944,245774727,389998116

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

%C Weigh transform of A050369.

%H Vaclav Kotesovec, <a href="/A307605/b307605.txt">Table of n, a(n) for n = 0..5000</a>

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

%F a(n) ~ ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(4 + 2*r)) * exp((2+r)/(1+r) * ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (2^(1/50) * sqrt(Pi*(2+r)) * n^((3 + r)/(4 + 2*r))), where r = A107311 is the root of the equation zeta(r) = 2. - _Vaclav Kotesovec_, Mar 18 2021

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 20*x^5 + 48*x^6 + 81*x^7 + 169*x^8 + 305*x^9 + ...

%t terms = 36; A[_] = 1; Do[A[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, A129373, A307604, A307606.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 18 2019