A102757 a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.

1, 4, 31, 352, 5233, 95836, 2080999, 52189096, 1482977857, 47053929268, 1648037039791, 63125834205424, 2624096058047281, 117620219281363852, 5653607876781921463, 290035426344483253816, 15814774125898034896129

Offset

0,2

Comments

Primes in this sequence include: a(2)=31, a(4)=5233. Semiprimes in this sequence include: a(1) = 2^2, a(6) = 31 * 67129, a(8) = 127 * 11676991. - Jonathan Vos Post, Mar 17 2005

Links

Seiichi Manyama, Table of n, a(n) for n = 0..377

Formula

E.g.f.: 1/(1-3x)*exp(x/(1-3x)).

E.g.f.: exp(3*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. [Paul D. Hanna, Nov 18 2011]

a(n) = 2*(3*n-1)*a(n-1) - 9*(n-1)^2*a(n-2). - Vaclav Kotesovec, Sep 29 2013

a(n) ~ (3*n)^(n+1/4)*exp(2*sqrt(n/3)-n-1/6)/sqrt(2) * (1 + 103/(144*sqrt(3*n))). - Vaclav Kotesovec, Sep 29 2013

Maple

seq(sum('binomial(k, i)^2*i!*3^i', 'i'=0..k), k=0..30);

Mathematica

f[n_] := Sum[k!*3^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *)
Range[0, 16]! CoefficientList[ Series[1/(1 - 3x)*Exp[x/(1 - 3x)], {x, 0, 16}], x] (* Robert G. Wilson v, Mar 16 2005 *)

Crossrefs

Cf. A002720, A025167, A102773.

Sequence in context: A129271 A136728 A321031 * A295254 A343832 A145561

Adjacent sequences: A102754 A102755 A102756 * A102758 A102759 A102760

Keyword

easy,nonn

Author

Miklos Kristof, Mar 16 2005

Extensions

More terms from Robert G. Wilson v, Mar 16 2005

Status

approved