A202831 Expansion of e.g.f.: exp(4*x/(1-5*x)) / sqrt(1-25*x^2).

1, 4, 81, 1444, 44521, 1397124, 58354321, 2574344644, 136043683281, 7657406908804, 489836445798001, 33351743794661604, 2504378700538997881, 199445618093659242244, 17189578072429077875121, 1564487078400498014277124, 152146464623361858013314721

Offset: 0

Author

Paul D. Hanna, Dec 25 2011

Keywords

• nonn

Examples

			E.g.f.: 1 + 4*x + 81*x^2/2! + 1444*x^3/3! + 44521*x^4/4! + 1397124*x^5/5! + ...
where A(x) = 1 + 2^2*x + 9^2*x^2/2! + 38^2*x^3/3! + 211^2*x^4/4! + 1182^2*x^5/5! + ... + A202832(n)^2*x^n/n! + ...
		

Links

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

Crossrefs

Cf. A202832, A202827, A202828, A202829, A202833, A202835, A202836.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-5*x))/Sqrt(1-25*x^2) ))); // G. C. Greubel, Jun 21 2022

Mathematica

CoefficientList[Series[Exp[4*x/(1-5*x)]/Sqrt[1-25*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

PARI

{a(n)=n!*polcoeff(exp(4*x/(1-5*x)+x*O(x^n))/sqrt(1-25*x^2+x*O(x^n)),n)}

PARI

{a(n)=n!^2*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)^2}

PARI

{a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))^2}

SageMath

def A202831_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-5*x))/sqrt(1-25*x^2) ).egf_to_ogf().list()
A202831_list(40) # G. C. Greubel, Jun 21 2022

Formula

a(n) = A202832(n)^2, where the e.g.f. of A202832 is exp(2*x + 5*x^2/2).

a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!) )^2.

a(n) ~ n^n*exp(4*sqrt(n/5)-2/5-n)*5^n/2. - Vaclav Kotesovec, May 23 2013

D-finite with recurrence: a(n) = (5*n-1)*a(n-1) + 5*(n-1)*(5*n-1)*a(n-2) - 125*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013