cp's OEIS Frontend

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

%I A081919 #35 Jan 30 2020 21:29:15
%S A081919 1,1,2,4,16,56,376,1912,17984,119296,1438336,11749376,172665472,
%T A081919 1674715264,29022277376,325841353216,6504163557376,82954203410432,
%U A081919 1874028623417344,26760916479840256,674914911967133696
%N A081919 E.g.f.: exp(x)/sqrt(1-x^2).
%C A081919 Binomial transform of aerated A001818 = 1, 0, 1, 0, 9, 0, 225, ... .
%C A081919 Number of perfect matchings in graph P_{2} X K_{n}. - _Andrew Howroyd_, Feb 28 2016
%H A081919 Robert Israel, <a href="/A081919/b081919.txt">Table of n, a(n) for n = 0..448</a>
%F A081919 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
%F A081919 a(n) ~ n^n * (exp(1)+(-1)^n*exp(-1)) / exp(n). - _Vaclav Kotesovec_, Feb 04 2014
%F A081919 a(n) = hyper3F0([1/2,-n/2,(1-n)/2],[],4). - _Peter Luschny_, Aug 21 2014
%F A081919 a(n) = sum_{k=0..floor(n/2)} ((2*k-1)!!)^2*binomial(n, 2*k). - _Andrew Howroyd_, Feb 28 2016
%F A081919 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
%p A081919 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):
%p A081919 map(f, [$0..25]); # _Robert Israel_, Feb 28 2016
%t A081919 CoefficientList[Series[E^x/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 04 2014 *)
%t A081919 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_ *)
%o A081919 (Sage)
%o A081919 A081919 = lambda n: hypergeometric([1/2,-n/2,(1-n)/2], [], 4)
%o A081919 [round(A081919(n).n()) for n in range(21)] # _Peter Luschny_, Aug 21 2014
%o A081919 (PARI) x='x+O('x^30); Vec(serlaplace(exp(x)/sqrt(1-x^2))) \\ _Michel Marcus_, Aug 21 2014
%Y A081919 Cf. A081920.
%K A081919 easy,nonn
%O A081919 0,3
%A A081919 _Paul Barry_, Apr 01 2003