A240957 - OEIS

%I #19 Aug 06 2014 06:02:05

%S 1,3,20,234,3944,86400,2324160,74062800,2726970624,113893395840,

%T 5319595814400,274730601277440,15544557784673280,956232958853652480,

%U 63540675378122342400,4535620918350762240000,346127227962539155292160,28120835253815298895380480,2423309442415144546546483200

%N G.f.: Sum_{n>=0} n^n * x^n * (3 + 2*n*x)^n / ((1 + n*x)*(1 + 2*n*x))^(n+1).

%H Vincenzo Librandi, <a href="/A240957/b240957.txt">Table of n, a(n) for n = 0..200</a>

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

%F 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

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

%e where

%e 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 +...

%t 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 *)

%o (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)}

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

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

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

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

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

%Y Cf. A240956, A240958, A240921.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 04 2014