A213193 O.g.f.: Sum_{n>=0} (4*n+1)^(4*n+1) * exp(-(4*n+1)^4*x) * x^n / n!.

1, 3124, 191757120, 49208861869440, 33030777426968816640, 45829974166034718596428800, 114009204539207742166715857223680, 462192193445890293982679086838571270144, 2851153321165202191241172917762717987236478976

Offset: 0

Author

Paul D. Hanna, Mar 01 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 + 3124*x + 191757120*x^2 + 49208861869440*x^3 +...
where
A(x) = exp(-x) + 5^5*x*exp(-5^4*x) + 9^9*exp(-9^4*x)*x^2/2! + 13^13*exp(-13^4*x)*x^3/3! + 17^17*exp(-17^4*x)*x^4/4! + 21^21*exp(-21^4*x)*x^5/5! +...
is a power series in x with integer coefficients.
		

Crossrefs

Cf. A222525, A220955, A221214, A217900, A007820, A242449.

Programs

Mathematica

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(4*k+1)^(4*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[4*n+1,n+k]*4^(n+k)*StirlingS2[n+k,n],{k,0,3*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

PARI

{a(n)=polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*exp(-(4*k+1)^4*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, (4*k+1)^(4*k+1)*x^k/(1+(4*k+1)^4*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)*(4*k+1)^(4*n+1))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} (4*k+1)^(4*k+1) * x^k / (1 + (4*k+1)^4*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (4*k+1)^(4*n+1).

a(n) ~ n^(3*n+1/2) * 2^(16*n+9/2) / (sqrt(2*Pi*(1-r)) * exp(3*n) * r^(n+1/4) * (4-r)^(3*n+1)), where r = -LambertW(-4*exp(-4)) = 0.0793096051271136564391... . - Vaclav Kotesovec, May 13 2014