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%I #9 Mar 03 2018 12:38:52
%S 1,3,41,967,32109,1373603,71889889,4448939407,317785091933,
%T 25731184562939,2328915407063705,233001395274021991,
%U 25533207271295208821,3041514682215417132371,391307447238067445930129,54074977650384006192679103,7988238906084714854241917421,1256227929202274469473017312811,209531162751200464078327250379657,36946198191974438054673167074349079
%N G.f.: Sum_{n>=0} (1 + (1 + 2*x)^n)^n / 2^(2*n+1).
%C Is there a closed-form expression for the terms of this sequence?
%F G.f. A(x) is given by:
%F (1) A(x) = Sum_{n>=0} 2 * (1 + (1 + 2*x)^n)^n / 4^(n+1).
%F (2) A(x) = Sum_{n>=0} 2 * (1 + 2*x)^(n^2) / (4 - (1 + 2*x)^n)^(n+1).
%e G.f.: A(x) = 1 + 3*x + 41*x^2 + 967*x^3 + 32109*x^4 + 1373603*x^5 + 71889889*x^6 + 4448939407*x^7 + 317785091933*x^8 + 25731184562939*x^9 + ...
%e such that
%e A(x) = 1/2 + (1 + (1+2*x))/2^3 + (1 + (1+2*x)^2)^2/2^5 + (1 + (1+2*x)^3)^3/2^7 + (1 + (1+2*x)^4)^4/2^9 + (1 + (1+2*x)^5)^5/2^11 + (1 + (1+2*x)^6)^6/2^13 + ...
%e Also, due to a series identity,
%e A(x) = 2/3 + 2*(1+2*x)/(4 - (1+2*x))^2 + 2*(1+2*x)^4/(4 - (1+2*x)^2)^3 + 2*(1+2*x)^9/(4 - (1+2*x)^3)^4 + 2*(1+2*x)^16/(4 - (1+2*x)^4)^5 + 2*(1+2*x)^25/(4 - (1+2*x)^5)^6 + 2*(1+2*x)^36/(4 - (1+2*x)^6)^7 + ...
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 03 2018
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