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A301581 G.f.: Sum_{n>=0} 2^n * ((1+x)^n - 1)^n.

%I A301581 #19 Oct 08 2020 04:24:59
%S A301581 1,2,16,232,4748,125440,4058312,155336672,6864980968,343995674280,
%T A301581 19270975801600,1193481831243584,80966964261458368,
%U A301581 5971270693661978816,475655179279901897536,40699219246551726635840,3722813577249648564213392,362519587815189751405383520,37442485808471509306691295808,4088344078912544484116541775616,470550859964811044524886252649760
%N A301581 G.f.: Sum_{n>=0} 2^n * ((1+x)^n - 1)^n.
%H A301581 Vaclav Kotesovec, <a href="/A301581/b301581.txt">Table of n, a(n) for n = 0..345</a>
%F A301581 G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) /(1 + 2*(1+x)^n)^(n+1).
%F A301581 a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 2*exp(1/r)) * r^2 = 5.90397654567002180048849478643459748789951071189960380570421209622277299181878..., where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.3270084259351451234078681822585067892123772641092056... - _Vaclav Kotesovec_, Aug 09 2018, updated Oct 08 2020
%e A301581 G.f.: A(x) = 1 + 2*x + 16*x^2 + 232*x^3 + 4748*x^4 + 125440*x^5 + 4058312*x^6 + 155336672*x^7 + 6864980968*x^8 + 343995674280*x^9 + ...
%e A301581 such that
%e A301581 A(x) = 1 + 2*((1+x)-1) + 4*((1+x)^2-1)^2 + 8*((1+x)^3-1)^3 + 16*((1+x)^4-1)^4 + 32*((1+x)^5-1)^5 + 64*((1+x)^6-1)^6 + 128*((1+x)^7-1)^7 + ...
%e A301581 Also,
%e A301581 A(x) = 1/3 + 2*(1+x)/(1 + 2*(1+x))^2 + 4*(1+x)^4/(1 + 2*(1+x)^2)^3 + 8*(1+x)^9/(1 + 2*(1+x)^3)^4 + 16*(1+x)^16/(1 + 2*(1+x)^4)^5 + 32*(1+x)^25/(1 + 2*(1+x)^5)^6 + ...
%t A301581 nmax = 20; CoefficientList[Series[1 + Sum[2^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 08 2020 *)
%o A301581 (PARI) {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 2^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
%o A301581 for(n=0,30,print1(a(n),", "))
%Y A301581 Cf. A122400, A195005, A301582, A301583, A301463.
%K A301581 nonn
%O A301581 0,2
%A A301581 _Paul D. Hanna_, Mar 24 2018