A098694 - OEIS

%I #42 Nov 27 2023 03:15:40

%S 1,2,48,34560,1393459200,5056584744960000,2422112183371431936000000,

%T 211155601241022491077587763200000000,

%U 4417964278440225627098723475313498521600000000000

%N Double-superfactorials: a(n) = Product_{k=1..n} (2k)!.

%C Hankel transform of double factorial numbers A001147. - _Paul Barry_, Jan 28 2008

%C Hankel transform of A112934(n+1). - _Paul Barry_, Dec 04 2009

%H Seiichi Manyama, <a href="/A098694/b098694.txt">Table of n, a(n) for n = 0..28</a>

%H C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

%F a(n) = Product_{k=0..n} (2*(k+1)*(2*k+1))^(n-k). - _Paul Barry_, Jan 28 2008

%F a(n) = A000178(n)*A057863(n)*A006125(n+1) = A121835(n)*A006125(n+1). - _Paul Barry_, Jan 28 2008

%F G.f.: G(0)/(2*x)-1/x, where G(k)= 1  + 1/(1 - 1/(1 + 1/(2*k+2)!/x/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 14 2013

%F a(n) ~ 2^(n^2+2*n+17/24) * n^(n^2+3*n/2+11/24) * Pi^((n+1)/2) / (A^(1/2) * exp(3*n^2/2+3*n/2-1/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Nov 13 2014

%F a(n) = A^(3/2)*2^(n^2+n-1/24)*Pi^(-n/2-1/4)*G(n+3/2)*G(n+2)/exp(1/8), where G(n) is the Barnes G-function and A is the Glaisher-Kinkelin constant. - _Ilya Gutkovskiy_, Dec 11 2016

%F a(n) = A000178(2*n + 1) / A168467(n). - _Vaclav Kotesovec_, Oct 28 2017

%t Table[Product[(2k)!,{k,1,n}],{n,0,10}] (* _Vaclav Kotesovec_, Nov 13 2014 *)

%o (PARI) a(n) = prod(k=1, n, (2*k)!); \\ _Michel Marcus_, Dec 11 2016

%o (Magma) [&*[ Factorial(2*k): k in [0..n] ]: n in [0..10]]; // _Vincenzo Librandi_, Dec 11 2016

%o (Python)

%o from math import prod

%o def A098694(n): return prod(((k+1)*((k<<1)+1)<<1)**(n-k) for k in range(1,n+1))<<n # _Chai Wah Wu_, Nov 26 2023

%Y Cf. A000178, A010050, A074962, A168467, A268504, A268505, A268506, A271946, A271947.

%K nonn

%O 0,2

%A _Ralf Stephan_, Sep 22 2004