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A244756
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a(n) = Sum_{k=0..n} C(n,k) * (2 + 3^k)^(n-k).
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3
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1, 4, 20, 136, 1424, 25504, 831680, 49526656, 5359464704, 1033951896064, 354410768092160, 213011725510260736, 224795751647646224384, 412813583857427719266304, 1323683536183041967893954560, 7361415226356149639592083685376, 71294465534894253722438522191806464
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: Sum_{n>=0} exp((2+3^n)*x) * x^n/n!.
O.g.f.: Sum_{n>=0} x^n/(1 - (2+3^n)*x)^(n+1).
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
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EXAMPLE
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E.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 136*x^3/3! + 1424*x^4/4! + 25504*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (2+3^0)^1 + (2+3^1)^0 = 4;
a(2) = (2+3^0)^2 + 2*(2+3^1)^1 + (2+3^2)^0 = 20;
a(3) = (2+3^0)^3 + 3*(2+3^1)^2 + 3*(2+3^2)^1 + (2+3^3)^0 = 136;
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; ...
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MATHEMATICA
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Table[Sum[Binomial[n, k] * (2 + 3^k)^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 25 2015 *)
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PROG
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(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (2 + 3^k)^(n-k) )}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. Sum_{n>=0} exp((2+3^n)*x)*x^n/n!" */
{a(n)=n!*polcoeff(sum(k=0, n, exp((2+3^k)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (2+3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2+3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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