A240958 G.f.: Sum_{n>=0} n^n * x^n * (1 + 2*n*x)^n / (1 + n*x + 2*n^2*x^2)^(n+1).

1, 1, 4, 30, 296, 3840, 60480, 1127280, 24240384, 590728320, 16090099200, 484387706880, 15971308784640, 572403619307520, 22155942961152000, 921115890645350400, 40935834850710159360, 1936630231160472207360, 97172886828612884889600, 5154401709528015200256000

Offset: 0

Author

Paul D. Hanna, Aug 04 2014

Keywords

• nonn

Comments

Generally, if Sum_{n>=0} n^n * x^n * (s + t*n*x)^n / (1 + s*n*x + t*n^2*x^2)^(n+1) = Sum_{n>=0} b(n)*x^n,

then b(n) = Sum_{k=0..n} (n-k)! * Stirling2(n, n-k) * binomial(n-k, k) * s^(n-2*k) * t^k.

Limit n->infinity (b(n)/n!)^(1/n) = s*r^2/(2*r-1) + (2*r-1)*r*t/((1-r)*s), where r is the root of the equation (r + (1-2*r)^2 * t/((1-r)*s^2)) * LambertW(-exp(-1/r)/r) = -1. - Vaclav Kotesovec, Aug 05 2014

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 296*x^4 + 3840*x^5 + 60480*x^6 +...
where
A(x) = 1 + x*(1+2*x)/(1+x+2*x^2)^2 + 2^2*x^2*(1+4*x)^2/(1+2*x+8*x^2)^3 + 3^3*x^3*(1+6*x)^3/(1+3*x+18*x^2)^4 + 4^4*x^4*(1+8*x)^4/(1+4*x+32*x^2)^5 + 5^5*x^5*(1+10*x)^5/(1+5*x+50*x^2)^6 +...
		

Links

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

Crossrefs

Cf. A240956, A240957, A240921.

Programs

Mathematica

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

PARI

/* By a general formula for o.g.f.: */
{a(n,s,t)=local(A=1); A=sum(m=0, n, m^m*x^m*(s + t*m*x)^m/(1 + s*m*x + t*m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n,1,2), ", "))

PARI

/* By a general formula for a(n): */
{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
{a(n,s,t)=sum(k=0, n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k, k)*s^(n-2*k)*t^k)}
for(n=0, 30, print1(a(n,1,2), ", "))

Formula

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

a(n) ~ c * d^n * n! / sqrt(n), where d = r^2/(2*r-1) + 2*(2*r-1)*r/(1-r) = 2.8672948250470036038473588196568091418984738141..., where r = 0.6842203847910787866923284795680321317882484098... is the root of the equation (r + 2*(1-2*r)^2/(1-r)) * LambertW(-exp(-1/r)/r) = -1, and c = 0.37767441309257908887250708986031213641309631613... . - Vaclav Kotesovec, Aug 05 2014