A081919 E.g.f.: exp(x)/sqrt(1-x^2).

1, 1, 2, 4, 16, 56, 376, 1912, 17984, 119296, 1438336, 11749376, 172665472, 1674715264, 29022277376, 325841353216, 6504163557376, 82954203410432, 1874028623417344, 26760916479840256, 674914911967133696

Offset: 0

Author

Paul Barry, Apr 01 2003

Keywords

• easy
• nonn

Comments

Binomial transform of aerated A001818 = 1, 0, 1, 0, 9, 0, 225, ... .

Number of perfect matchings in graph P_{2} X K_{n}. - Andrew Howroyd, Feb 28 2016

Links

• Robert Israel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A081920.

Programs

Maple

f:= gfun:-rectoproc({-a(n) +a(n-1) +(n-1)^2*a(n-2) -(n-1)*(n-2)*a(n-3)=0, a(0) = 1, a(1)=1,a(2)=2},a(n),remember):
map(f, [$0..25]); # Robert Israel, Feb 28 2016

Mathematica

CoefficientList[Series[E^x/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 04 2014 *)
a[n_] := Sum[((2k-1)!!)^2 Binomial[n, 2k], {k, 0, n/2}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(x)/sqrt(1-x^2))) \\ Michel Marcus, Aug 21 2014

Sage

A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)
[round(A081919(n).n()) for n in range(21)] # Peter Luschny, Aug 21 2014

Formula

D-finite with recurrence: -a(n) +a(n-1) +(n-1)^2*a(n-2) -(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 09 2012

a(n) ~ n^n * (exp(1)+(-1)^n*exp(-1)) / exp(n). - Vaclav Kotesovec, Feb 04 2014

a(n) = hyper3F0([1/2,-n/2,(1-n)/2],[],4). - Peter Luschny, Aug 21 2014

a(n) = sum_{k=0..floor(n/2)} ((2*k-1)!!)^2*binomial(n, 2*k). - Andrew Howroyd, Feb 28 2016

E.g.f. A(x) satisfies (1-x^2)*A'(x) - (1+x-x^2)*A(x) = 0, from which R. J. Mathar's recurrence follows. - Robert Israel, Feb 28 2016