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A222525 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n) * exp(-(2*n+1)^2*x) * x^n / n!. 4
1, 8, 232, 12160, 929376, 93590784, 11709432064, 1751777730560, 305065968649728, 60623947402670080, 13538933075023376384, 3356940619048979988480, 915040828127405123420160, 271974910674004076827115520, 87543520972441760055430348800, 30337462571518006406505729884160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

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

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

a(n) ~ 2^(4*n) * n^(n-1/2) / (sqrt(Pi*r*(1-r)) * exp(n) * (r*(2-r))^n), where r = -LambertW(-2*exp(-2)) = 0.4063757399599... (see A226775 = -r). - Vaclav Kotesovec, May 13 2014

EXAMPLE

O.g.f.: A(x) = 1 + 8*x + 232*x^2 + 12160*x^3 + 929376*x^4 + 93590784*x^5 +...

where

A(x) = exp(-x) + 3^2*exp(-3^2*x)*x + 5^4*exp(-5^2*x)*x^2/2! + 7^6*exp(-7^2*x)*x^3/3! + 9^8*exp(-9^2*x)*x^4/4! + 11^10*exp(-11^2*x)*x^5/5! +...

is a power series in x with integer coefficients.

MATHEMATICA

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

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

PROG

(PARI) {a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k)*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)*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))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A220955, A217900, A007820, A242373.

Sequence in context: A300209 A300798 A240299 * A214351 A303450 A301437

Adjacent sequences:  A222522 A222523 A222524 * A222526 A222527 A222528

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 24 2013

STATUS

approved

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Last modified January 28 01:55 EST 2022. Contains 350654 sequences. (Running on oeis4.)