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%I #39 Dec 18 2022 02:33:33
%S 1,4,40,640,14080,394240,13404160,536166400,24663654400,1282510028800,
%T 74385581670400,4760677226905600,333247405883392000,
%U 25326802847137792000,2076797833465298944000,182758209344946307072000,17179271678424952864768000,1717927167842495286476800000
%N Sextuple factorial numbers: Product_{k=0..n-1} (6*k+4).
%H G. C. Greubel, <a href="/A049308/b049308.txt">Table of n, a(n) for n = 0..340</a>
%F E.g.f.: (1-6*x)^(-2/3).
%F a(n) = 2^n*A008544(n).
%F G.f.: 1/(1-4*x/(1-6*x/(1-10*x/(1-12*x/(1-16*x/(1-18*x/(1-22*x/(1-24*x/(1-28*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F a(n) = (-2)^n*Sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012
%F G.f.: ( 1 - 1/Q(0) )/x/2 where Q(k) = 1 - x*(6*k-2)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013
%F D-finite with recurrence: a(n) = 2*(3*n-1)*a(n-1). - _R. J. Mathar_, Jan 17 2020
%F From _G. C. Greubel_, Mar 29 2022: (Start)
%F a(n) = 6^n * Pochhammer(n, 2/3).
%F G.f.: Hypergeometric2F0([1, 2/3], [], 6*x). (End)
%F Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(2/3) - Gamma(2/3, 1/6))/6^(1/3). - _Amiram Eldar_, Dec 18 2022
%t Table[6^n*Pochhammer[2/3, n], {n,0,30}] (* _G. C. Greubel_, Mar 29 2022 *)
%o (Magma) [n le 2 select 4^(n-1) else 2*(3*n-1)*Self(n-1): n in [1..30]]; // _G. C. Greubel_, Mar 29 2022
%o (Sage) [6^n*rising_factorial(2/3,n) for n in (0..30)] # _G. C. Greubel_, Mar 29 2022
%o (PARI) a(n) = prod(k=0, n-1, 6*k+4); \\ _Michel Marcus_, Mar 30 2022
%Y Cf. A008543, A008544, A011781, A048994.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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