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A098694 Double-superfactorials: a(n) = Product_{k=1..n} (2k)!. 18
1, 2, 48, 34560, 1393459200, 5056584744960000, 2422112183371431936000000, 211155601241022491077587763200000000, 4417964278440225627098723475313498521600000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hankel transform of double factorial numbers A001147. - Paul Barry, Jan 28 2008

Hankel transform of A112934(n+1). - Paul Barry, Dec 04 2009

LINKS

Table of n, a(n) for n=0..8.

C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

FORMULA

a(n) = Product_{k=0..n} (2*(k+1)*(2*k+1))^(n-k). - Paul Barry, Jan 28 2008

a(n) = A000178(n)*A057863(n)*A006125(n+1) = A121835(n)*A006125(n+1). - Paul Barry, Jan 28 2008

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

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

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

a(n) = A000178(2*n + 1) / A168467(n). - Vaclav Kotesovec, Oct 28 2017

MATHEMATICA

Table[Product[(2k)!, {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 13 2014 *)

PROG

(PARI) a(n) = prod(k=1, n, (2*k)!); \\ Michel Marcus, Dec 11 2016

(MAGMA) [&*[ Factorial(2*k): k in [0..n] ]: n in [0..10]]; // Vincenzo Librandi, Dec 11 2016

CROSSREFS

Cf. A000178, A010050, A074962, A268504, A268505, A268506, A271946, A271947.

Sequence in context: A203311 A295177 * A137592 A203778 A203305 A191954

Adjacent sequences:  A098691 A098692 A098693 * A098695 A098696 A098697

KEYWORD

nonn

AUTHOR

Ralf Stephan, Sep 22 2004

STATUS

approved

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Last modified February 19 05:12 EST 2018. Contains 299330 sequences. (Running on oeis4.)