A202827 Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).

1, 4, 25, 196, 1849, 20164, 249001, 3422500, 51739249, 851822596, 15155825881, 289527934084, 5906625426025, 128089110981316, 2940882813228649, 71239270847432164, 1815115761586307041, 48511703775281296900, 1356708799439194070809, 39615996090901693902916

Offset

0,2

Links

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

Formula

a(n) = A005425(n)^2, where the e.g.f. of A005425 is exp(2*x + x^2/2).

a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*n!/((n-2*k)!*k!) )^2. [From formula by Huajun Huang in A005425]

a(n) ~ n^n*exp(4*sqrt(n)-2-n)/2 * (1+5/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013

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

Example

E.g.f.: A(x) = 1 + 4*x + 25*x^2/2! + 196*x^3/3! + 1849*x^4/4! +...
where A(x) = 1 + 2^2*x + 5^2*x^2/2! + 14^2*x^3/3! + 43^2*x^4/4! +...+ A005425(n)^2*x^n/n! +...

Mathematica

With[{nn=20}, CoefficientList[Series[Exp[(4x)/(1-x)]/Sqrt[1-x^2], {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Dec 31 2011 *)

Prog

(PARI) {a(n)=n!*polcoeff(exp(4*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n\2, 2^(n-3*k)*n!/((n-2*k)!*k!))^2}
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 21 2022
(SageMath)
def A202827_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202827_list(40) # G. C. Greubel, Jun 21 2022

Crossrefs

Cf. A005425, A202828, A202829, A202831, A202833, A202835, A202836.

Sequence in context: A206179 A151342 A001246 * A065735 A212694 A182953

Adjacent sequences: A202824 A202825 A202826 * A202828 A202829 A202830

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Status

approved