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A194353 G.f.: exp( Sum_{n>=1} (x^n/n)/sqrt(1 - 4*x^n) ).

%I A194353 #14 May 10 2021 04:16:06
%S A194353 1,1,3,9,32,114,431,1633,6306,24466,95604,374920,1475731,5822941,
%T A194353 23027817,91225983,361945480,1437855016,5718242653,22762417519,
%U A194353 90684186378,361540944078,1442313708600,5757140994990,22991764612516,91861397186492
%N A194353 G.f.: exp( Sum_{n>=1} (x^n/n)/sqrt(1 - 4*x^n) ).
%H A194353 Vaclav Kotesovec, <a href="/A194353/b194353.txt">Table of n, a(n) for n = 0..1000</a>
%F A194353 G.f.: Product_{n>=1} 1/(1-x^n)^binomial(2*n-2,n-1).
%F A194353 a(n) ~ 2^(2*n-1) * exp(3*n^(1/3)/4 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} 1/(k * 4^k * sqrt(1 - 4^(1-k))) = 0.0426953980609366317948752143053832860841846208508627487697950... - _Vaclav Kotesovec_, May 10 2021
%e A194353 G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 114*x^5 + 431*x^6 +...
%e A194353 where
%e A194353 log(A(x)) = x/sqrt(1-4*x) + (x^2/2)/sqrt(1-4*x^2) + (x^3/3)/sqrt(1-4*x^3) +...
%e A194353 Other expansions:
%e A194353 log(A(x)) = x*(1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...)
%e A194353 + x^2/2*(1 + 2*x^2 + 6*x^4 + 20*x^6 + 70*x^8 + 252*x^10 +...)
%e A194353 + x^3/3*(1 + 2*x^3 + 6*x^6 + 20*x^9 + 70*x^12 + 252*x^15 +...) +...
%e A194353 log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 85*x^4/4 + 351*x^5/5 + 1535*x^6/6 + 6469*x^7/7 +...
%e A194353 1/A(x) = (1-x)*(1-x^2)^2*(1-x^3)^6*(1-x^4)^20*(1-x^5)^70*(1-x^6)^252*...
%t A194353 nmax = 25; CoefficientList[Series[Exp[Sum[x^j/j/Sqrt[1 - 4*x^j], {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 10 2021 *)
%t A194353 nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2*k - 2, k - 1], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 10 2021 *)
%o A194353 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,(x^m/m)/sqrt(1-4*x^m+x*O(x^n)))),n)}
%Y A194353 Cf. A088327, A344108.
%K A194353 nonn
%O A194353 0,3
%A A194353 _Paul D. Hanna_, Aug 23 2011