|
%I #19 Jan 12 2017 04:40:24
%S 1,2,6,40,210,2016,13860,164736,1351350,18670080,174594420,2708858880,
%T 28109701620,479259648000,5421156741000,100069414502400,
%U 1218404977539750,24087296714342400,312723944235202500,6566957735804928000,90252130306279441500,2000107698962300928000
%N a(n) = (2^floor(n+n/2)/sqrt(Pi)^mod(n+1,2))*Gamma(n+1/2)/Gamma(n/2+1).
%H G. C. Greubel, <a href="/A264152/b264152.txt">Table of n, a(n) for n = 0..565</a>
%F a(n) = A001813(n)/A006882(n).
%F a(n) = A000079(n)*A006882(2*n-1)/A006882(n).
%F E.g.f.: 2F2(1/4,3/4;1/2,1;8*x^2) + 2*x*2F2(3/4,5/4;3/2,3/2;8*x^2). - _Benedict W. J. Irwin_, Aug 15 2016
%F a(n) ~ 2^(2*n) * n^((n-1)/2) * exp(-n/2) * (2/Pi)^((1+(-1)^n)/4). - _Ilya Gutkovskiy_, Aug 15 2016
%p a := n -> (2^floor(n+n/2)/sqrt(Pi)^modp(n+1,2))*GAMMA(n+1/2)/GAMMA(n/2+1):
%p seq(a(n), n=0..21);
%t Table[CoefficientList[Series[HypergeometricPFQ[{1/4, 3/4}, {1/2, 1}, 8 x^2] +
%t 2 x HypergeometricPFQ[{3/4, 5/4}, {3/2, 3/2}, 8 x^2], {x, 0, 20}], x][[n]] (n - 1)!, {n, 1, 20}] (* _Benedict W. J. Irwin_, Aug 15 2016 *)
%t Table[(2^Floor[n + n/2]/Sqrt[Pi]^Mod[n + 1, 2])*Gamma[n + 1/2]/Gamma[n/2 + 1], {n, 0, 20}] (* _Benedict W. J. Irwin_, Aug 15 2016 *)
%o (Sage)
%o a = lambda n: (rising_factorial(1/2, n) // n.multifactorial(2)) << 2*n
%o [a(n) for n in (0..21)]
%Y Cf. A000079, A001813, A006882.
%K nonn
%O 0,2
%A _Peter Luschny_, Nov 06 2015
|
