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%I #19 Feb 04 2017 21:22:22
%S 1,3,13,87,985,19563,697573,44195007,4985202865,987432857043,
%T 344306650353853,209169206074748967,222262777197258910345,
%U 409907753371580011362363,1317924525238880964004945813,7341603216747343890845790989967,71176841502529490992224798115792225
%N a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k).
%H G. C. Greubel, <a href="/A244755/b244755.txt">Table of n, a(n) for n = 0..90</a>
%F E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * x^n/n!.
%F O.g.f.: Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1).
%F a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - _Vaclav Kotesovec_, Jan 25 2015
%e E.g.f.: A(x) = 1 + 3*x + 13*x^2/2! + 87*x^3/3! + 985*x^4/4! + 19563*x^5/5! +...
%e ILLUSTRATION OF INITIAL TERMS:
%e a(1) = (1+3^0)^1 + (1+3^1)^0 = 3;
%e a(2) = (1+3^0)^2 + 2*(1+3^1)^1 + (1+3^2)^0 = 13;
%e a(3) = (1+3^0)^3 + 3*(1+3^1)^2 + 3*(1+3^2)^1 + (1+3^3)^0 = 87;
%e a(4) = (1+3^0)^4 + 4*(1+3^1)^3 + 6*(1+3^2)^2 + 4*(1+3^3)^1 + (1+3^4)^0 = 985; ...
%t Table[Sum[Binomial[n,k] * (1 + 3^k)^(n-k),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 25 2015 *)
%o (PARI) {a(n) = sum(k=0,n,binomial(n,k) * (1 + 3^k)^(n-k) )}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*x^n/n!" */
%o {a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*x^k/k!), n)}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1): */
%o {a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A244754, A244756, A244760, A135079, A243918.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 05 2014
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