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%I #46 Dec 18 2022 02:28:03
%S 1,2,16,224,4480,116480,3727360,141639680,6232145920,311607296000,
%T 17450008576000,1081900531712000,73569236156416000,
%U 5444123475574784000,435529878045982720000,37455569511954513920000,3445912395099815280640000,337699414719781897502720000
%N Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).
%H G. C. Greubel, <a href="/A047657/b047657.txt">Table of n, a(n) for n = 0..340</a>
%F E.g.f.: (1-6*x)^(-1/3).
%F a(n) = 2^n*A007559(n).
%F a(n) = A084941(n)/A000142(n)*A000079(n) = 6^n*Pochhammer(1/3, n) = 1/2*6^n*Gamma(n+1/3)*sqrt(3)*Gamma(2/3)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
%F Let b(n) = b(n-1) + 6; then a(n) = b(n)*a(n-1). - _Roger L. Bagula_, Sep 17 2008
%F G.f.: 1/(1-2*x/(1-6*x/(1-8*x/(1-12*x/(1-14*x/(1-18*x/(1-20*x/(1-24*x/(1-26*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F a(n) = (-4)^n*Sum_{k=0..n} (3/2)^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/G(0) where G(k) = 1 - x*(6*k+2)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 23 2013
%F D-finite with recurrence: a(n) +2*(-3*n+2)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(1/3) - Gamma(1/3, 1/6))/6^(2/3). - _Amiram Eldar_, Dec 18 2022
%p a:= n->product(6*j+2, j=0..n-1); seq(a(n), n=0..20); # _G. C. Greubel_, Aug 18 2019
%t b[1]=2; b[n_]:= b[n] = b[n-1] +6; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n,0,20}] (* _Roger L. Bagula_, Sep 17 2008 *)
%t FoldList[Times,1,6*Range[0,20]+2] (* _Harvey P. Dale_, Aug 06 2013 *)
%t Table[6^n*Pochhammer[1/3, n], {n,0,20}] (* _G. C. Greubel_, Aug 18 2019 *)
%o (PARI) vector(20, n, n--; prod(k=0, n-1, 6*k+2)) \\ _G. C. Greubel_, Aug 18 2019
%o (Magma) [1] cat [(&*[6*k+2: k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 18 2019
%o (Sage) [product(6*k+2 for k in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Aug 18 2019
%o (GAP) List([0..20], n-> Product([0..n-1], k-> 6*k+2) ); # _G. C. Greubel_, Aug 18 2019
%Y Cf. A007559, A008542, A011781.
%Y Cf. A000165, A008544, A001813, A047055, A084947, A084948, A084949.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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