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A244760 a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k) * 2^k. 3
1, 4, 24, 232, 3840, 111904, 5785344, 529662592, 85449338880, 24204383609344, 11986829259362304, 10361640102119729152, 15589910824599107174400, 40815393380277274447519744, 185575767151388880816233447424, 1465910356757779350231777997914112 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..89

FORMULA

E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * (2*x)^n/n!.

O.g.f.: Sum_{n>=0} (2*x)^n/(1 - (1+3^n)*x)^(n+1).

a(n) ~ c * 3^(n^2/4) * 2^((3*n+1)/2) / sqrt(Pi*n), where c = sum_{k=-inf..+inf} 1/(3^(k^2) * 2^k) = 1.88621563508001862566062... if n is even, and c = sum_{k=-inf..+inf} 1/(3^((k+1/2)^2) * 2^(k+1/2)) = 1.88659407336643412717014... if n is odd. - Vaclav Kotesovec, Jan 25 2015

EXAMPLE

E.g.f.: A(x) = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3840*x^4/4! + 111904*x^5/5! +...

ILLUSTRATION OF INITIAL TERMS:

a(1) = (1+3^0)^1 + (1+3^1)^0*2 = 4;

a(2) = (1+3^0)^2 + 2*(1+3^1)^1*2 + (1+3^2)^0*2^2 = 24;

a(3) = (1+3^0)^3 + 3*(1+3^1)^2*2 + 3*(1+3^2)^1*2^2 + (1+3^3)^0*2^3 = 232;

a(4) = (1+3^0)^4 + 4*(1+3^1)^3*2 + 6*(1+3^2)^2*2^2 + 4*(1+3^3)^1*2^3 + (1+3^4)^0*2^4 = 3480; ...

MATHEMATICA

Table[Sum[Binomial[n, k] * (1 + 3^k)^(n-k) * 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 25 2015 *)

PROG

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (1 + 3^k)^(n-k)*2^k )}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*(2*x)^n/n! */

{a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*(2*x)^k/k!), n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* O.g.f. Sum_{n>=0} (2*x)^n/(1 - (1+3^n)*x)^(n+1): */

{a(n)=polcoeff(sum(k=0, n, (2*x)^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A244755, A244756, A244754, A243918.

Sequence in context: A318005 A224800 A234012 * A240403 A074598 A307826

Adjacent sequences:  A244757 A244758 A244759 * A244761 A244762 A244763

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 05 2014

STATUS

approved

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Last modified January 21 10:26 EST 2020. Contains 331105 sequences. (Running on oeis4.)