0,2

Vincenzo Librandi, Table of n, a(n) for n = 0..200

a(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * 3^(n-2*k) * 2^k.

a(n) ~ c * d^n * n! / sqrt(n), where d = 3*r^2/(2*r-1) + 2*(2*r-1)*r/(3*(1-r)) = 4.927267464017203368228591159442769988364645445182..., where r = 0.8093509687086163798199326301917112747442352555652682... is the root of the equation (r + 2*(1-2*r)^2/(9*(1-r))) * LambertW(-exp(-1/r)/r) = -1, and c = 0.546345652881951027770637598235474648132398514044679... . - Vaclav Kotesovec, Aug 05 2014

O.g.f.: A(x) = 1 + 3*x + 20*x^2 + 234*x^3 + 3944*x^4 + 86400*x^5 +...

where

A(x) = 1 + x*(3+2*x)/((1+x)*(1+2*x))^2 + 2^2*x^2*(3+4*x)^2/((1+2*x)*(1+4*x))^3 + 3^3*x^3*(3+6*x)^3/((1+3*x)*(1+6*x))^4 + 4^4*x^4*(3+8*x)^4/((1+4*x)*(1+8*x))^5 + 5^5*x^5*(3+10*x)^5/((1+5*x)*(1+10*x))^6 +...

Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k, k] * 3^(n-2*k) * 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 05 2014 *)

(PARI) {a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(3+2*m*x)^m/((1 + m*x)*(1+2*m*x) +x*O(x^n))^(m+1)); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* From formula for a(n): */

{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}

{a(n)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k)*3^(n-2*k)*2^k)}

for(n=0, 30, print1(a(n), ", "))

Cf. A240956, A240958, A240921.

Sequence in context: A108527 A194972 A294603 * A335871 A195135 A201148

Adjacent sequences: A240954 A240955 A240956 * A240958 A240959 A240960

Paul D. Hanna, Aug 04 2014

approved