A240957 G.f.: Sum_{n>=0} n^n * x^n * (3 + 2*n*x)^n / ((1 + n*x)*(1 + 2*n*x))^(n+1).
1, 3, 20, 234, 3944, 86400, 2324160, 74062800, 2726970624, 113893395840, 5319595814400, 274730601277440, 15544557784673280, 956232958853652480, 63540675378122342400, 4535620918350762240000, 346127227962539155292160, 28120835253815298895380480, 2423309442415144546546483200
Offset: 0
Keywords
Examples
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 +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Mathematica
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), ", "))
Formula
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