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A244756 a(n) = Sum_{k=0..n} C(n,k) * (2 + 3^k)^(n-k).

%I A244756 #17 Feb 15 2017 03:04:47
%S A244756 1,4,20,136,1424,25504,831680,49526656,5359464704,1033951896064,
%T A244756 354410768092160,213011725510260736,224795751647646224384,
%U A244756 412813583857427719266304,1323683536183041967893954560,7361415226356149639592083685376,71294465534894253722438522191806464
%N A244756 a(n) = Sum_{k=0..n} C(n,k) * (2 + 3^k)^(n-k).
%H A244756 G. C. Greubel, <a href="/A244756/b244756.txt">Table of n, a(n) for n = 0..90</a>
%F A244756 E.g.f.: Sum_{n>=0} exp((2+3^n)*x) * x^n/n!.
%F A244756 O.g.f.: Sum_{n>=0} x^n/(1 - (2+3^n)*x)^(n+1).
%F A244756 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 A244756 E.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 136*x^3/3! + 1424*x^4/4! + 25504*x^5/5! +...
%e A244756 ILLUSTRATION OF INITIAL TERMS:
%e A244756 a(1) = (2+3^0)^1 + (2+3^1)^0 = 4;
%e A244756 a(2) = (2+3^0)^2 + 2*(2+3^1)^1 + (2+3^2)^0 = 20;
%e A244756 a(3) = (2+3^0)^3 + 3*(2+3^1)^2 + 3*(2+3^2)^1 + (2+3^3)^0 = 136;
%e A244756 a(4) = (2+3^0)^4 + 4*(2+3^1)^3 + 6*(2+3^2)^2 + 4*(2+3^3)^1 + (2+3^4)^0 = 1424; ...
%t A244756 Table[Sum[Binomial[n,k] * (2 + 3^k)^(n-k),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 25 2015 *)
%o A244756 (PARI) {a(n) = sum(k=0,n,binomial(n,k) * (2 + 3^k)^(n-k) )}
%o A244756 for(n=0,25,print1(a(n),", "))
%o A244756 (PARI) /* E.g.f. Sum_{n>=0} exp((2+3^n)*x)*x^n/n!" */
%o A244756 {a(n)=n!*polcoeff(sum(k=0, n, exp((2+3^k)*x +x*O(x^n))*x^k/k!), n)}
%o A244756 for(n=0,25,print1(a(n),", "))
%o A244756 (PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (2+3^n)*x)^(n+1): */
%o A244756 {a(n)=polcoeff(sum(k=0, n, x^k/(1-(2+3^k)*x +x*O(x^n))^(k+1)), n)}
%o A244756 for(n=0,25,print1(a(n),", "))
%Y A244756 Cf. A244755, A244760, A244754, A243918.
%K A244756 nonn
%O A244756 0,2
%A A244756 _Paul D. Hanna_, Jul 05 2014