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A244756 a(n) = Sum_{k=0..n} C(n,k) * (2 + 3^k)^(n-k). 3
1, 4, 20, 136, 1424, 25504, 831680, 49526656, 5359464704, 1033951896064, 354410768092160, 213011725510260736, 224795751647646224384, 412813583857427719266304, 1323683536183041967893954560, 7361415226356149639592083685376, 71294465534894253722438522191806464 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..90

FORMULA

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

EXAMPLE

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; ...

MATHEMATICA

Table[Sum[Binomial[n, k] * (2 + 3^k)^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 25 2015 *)

PROG

(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), ", "))

CROSSREFS

Cf. A244755, A244760, A244754, A243918.

Sequence in context: A141716 A129102 A129099 * A292932 A187116 A098541

Adjacent sequences:  A244753 A244754 A244755 * A244757 A244758 A244759

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 05 2014

STATUS

approved

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Last modified February 18 09:39 EST 2020. Contains 332011 sequences. (Running on oeis4.)