A202878 Expansion of e.g.f.: exp(16*x/(1-x)) / sqrt(1-x^2).

1, 16, 289, 5776, 126025, 2972176, 75186241, 2027520784, 57988974481, 1751546371600, 55668326576641, 1855807478279056, 64713593898036889, 2354701531657512976, 89209297718289390625, 3512141211682081889296, 143435878498076017059361

Offset: 0

Author

Paul D. Hanna, Dec 25 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 16*x + 289*x^2/2! + 5776*x^3/3! + 126025*x^4/4! + ...
where A(x) = 1 + 4^2*x + 17^2*x^2/2! + 76^2*x^3/3! + 355^2*x^4/4! + 1724^2*x^5/5! + ... + A202879(n)^2*x^n/n! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A202879, A202827, A202828, A202829, A202831, A202835, A202836, A202837.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(16*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022

Mathematica

CoefficientList[Series[Exp[16*x/(1-x)]/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

PARI

{a(n)=n!*polcoeff(exp(16*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

PARI

{a(n)=n!^2*polcoeff(exp(4*x+x^2/2+x*O(x^n)),n)^2}

SageMath

def A202878_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(16*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202878_list(40) # G. C. Greubel, Jun 22 2022

Formula

a(n) = A202879(n)^2, where the e.g.f. of A202879 is exp(4*x + x^2/2).

a(n) = ( Sum_{k=0..floor(n/2)} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.

a(n) ~ n^n*exp(8*sqrt(n)-8-n)/2 * (1+22/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013

D-finite with recurrence: a(n) = (n+15)*a(n-1) + (n-1)*(n+15)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013