A220955 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n+1) * exp(-(2*n+1)^2*x) * x^n / n!.

1, 26, 1320, 99288, 9901920, 1230768704, 183260197120, 31800433551744, 6301891570411008, 1404224096732154880, 347532097449969496064, 94584986134590717358080, 28076463606243146379018240, 9027122730610037995425792000, 3125219575155651450096795648000

Offset: 0

Author

Paul D. Hanna, Feb 27 2013

Keywords

• nonn

Comments

From Vaclav Kotesovec, May 13 2014: (Start)

Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).

a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).

(End)

Examples

			O.g.f.: A(x) = 1 + 26*x + 1320*x^2 + 99288*x^3 + 9901920*x^4 +...
where A(x) = exp(-x) + 3^3*exp(-3^2*x)*x + 5^5*exp(-5^2*x)*x^2/2! + 7^7*exp(-7^2*x)*x^3/3! + 9^9*exp(-9^2*x)*x^4/4! + 11^11*exp(-11^2*x)*x^5/5! +...
is a power series in x with integer coefficients.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..290

Crossrefs

Cf. A222525, A221214, A213193, A217900, A007820, A242449.

Programs

Mathematica

Table[1/n! * Sum[(-1)^(n-k)*Binomial[n,k] * (2*k+1)^(2*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[2*n+1,n+k]*2^(n+k)*StirlingS2[n+k,n],{k,0,n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

PARI

{a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n+1))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k+1) * x^k / (1 + (2*k+1)^2*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n+1).

a(n) = Sum_{k=0..n+1} 2^(n+k) * binomial(2*n+1,n+k) * stirling2(n+k,n). - Vaclav Kotesovec, May 13 2014

a(n) ~ n^(n+1/2) * 2^(4*n+5/2) / (sqrt(2*Pi*(1-r)) * exp(n) * r^(n+1/2) * (2-r)^(n+1)), where r = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775 = -r) . - Vaclav Kotesovec, May 13 2014