%I #8 Aug 09 2018 04:42:15
%S 1,3,54,1494,58248,2921346,179119836,12981530772,1085678924472,
%T 102911898062376,10903265835968178,1276820000444309958,
%U 163765996498409795046,22831772648676453195534,3437850237146915605162722,555999871854064840852040190,96123855383854022518187481072,17690662829477220311541393099324,3453143617992367493730150308612370
%N G.f.: Sum_{n>=0} ( (1+x)^n - 1 )^n * 3^n / (4 - 3*(1+x)^n)^(n+1).
%C The following identities hold for |y| <= 1 and fixed real k > 0:
%C (C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
%C (C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1).
%C This sequence is an example of (C2) when y = 1+x and k = 3.
%H Vaclav Kotesovec, <a href="/A317663/b317663.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 3^n / (4 - 3*(1+x)^n)^(n+1).
%F (2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 3^n / (4 + 3*(1+x)^n)^(n+1).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = 11.154788564351081167494585241180262193438722530344791058752757035461192417... and c = 0.321897864665202841967234839159770976446040882710871129852558... - _Vaclav Kotesovec_, Aug 09 2018
%e G.f.: A(x) = 1 + 3*x + 54*x^2 + 1494*x^3 + 58248*x^4 + 2921346*x^5 + 179119836*x^6 + 12981530772*x^7 + 1085678924472*x^8 + ...
%e such that
%e A(x) = 1 + ((1+x) - 1)*3/(4 - 3*(1+x))^2 + ((1+x)^2 - 1)^2*3^2/(4 - 3*(1+x)^2)^3 + ((1+x)^3 - 1)^3*3^3/(4 - 3*(1+x)^3)^4 + ((1+x)^4 - 1)^4*3^4/(4 - 3*(1+x)^4)^5 + ((1+x)^5 - 1)^5*3^5/(4 - 3*(1+x)^5)^6 + ((1+x)^6 - 1)^6*3^6/(4 - 3*(1+x)^6)^7 + ...
%e Also,
%e A(x) = 1/7 + ((1+x) + 1)*3/(4 + 3*(1+x))^2 + ((1+x)^2 + 1)^2*3^2/(4 + 3*(1+x)^2)^3 + ((1+x)^3 + 1)^3*3^3/(4 + 3*(1+x)^3)^4 + ((1+x)^4 + 1)^4*3^4/(4 + 3*(1+x)^4)^5 + ((1+x)^5 + 1)^5*3^5/(4 + 3*(1+x)^5)^6 + ((1+x)^6 + 1)^6*3^6/(4 + 3*(1+x)^6)^7 + ...
%o (PARI) {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 3^m / (4 - 3*(1+x)^m +x*O(x^n) )^(m+1) ); ;polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A302598, A317662, A317664, A302615.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 03 2018