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A244760 a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k) * 2^k. 3

%I

%S 1,4,24,232,3840,111904,5785344,529662592,85449338880,24204383609344,

%T 11986829259362304,10361640102119729152,15589910824599107174400,

%U 40815393380277274447519744,185575767151388880816233447424,1465910356757779350231777997914112

%N a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k) * 2^k.

%H G. C. Greubel, <a href="/A244760/b244760.txt">Table of n, a(n) for n = 0..89</a>

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

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

%F 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

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

%e ILLUSTRATION OF INITIAL TERMS:

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

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

%e 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;

%e 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; ...

%t Table[Sum[Binomial[n,k] * (1 + 3^k)^(n-k) * 2^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)*2^k )}

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

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

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

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

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

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

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

%Y Cf. A244755, A244756, A244754, A243918.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 05 2014

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Last modified February 26 07:17 EST 2020. Contains 332277 sequences. (Running on oeis4.)