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A008543 Sextuple factorial numbers: Product_{k=0..n-1} (6*k + 5).

%I A008543 #76 Feb 01 2023 12:27:15
%S A008543 1,5,55,935,21505,623645,21827575,894930575,42061737025,2229272062325,
%T A008543 131527051677175,8549258359016375,606997343490162625,
%U A008543 46738795448742522125,3879320022245629336375,345259481979861010937375,32799650788086796039050625,3312764729596766399944113125
%N A008543 Sextuple factorial numbers: Product_{k=0..n-1} (6*k + 5).
%H A008543 Vincenzo Librandi, <a href="/A008543/b008543.txt">Table of n, a(n) for n = 0..300</a>
%H A008543 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%F A008543 a(n) = 5*A034787(n) = (6*n-1)(!^6), n >= 1, a(0) := 1.
%F A008543 E.g.f.: (1 - 6*x)^(-5/6).
%F A008543 a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(5/6)^-1*n^(1/3)*6^n*e^-n*n^n*(1 + (1/72)*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F A008543 G.f.: 1/(1-5x/(1-6x/(1-11x/(1-12x/(1-17x/(1-18x/(1-23x/(1-24x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F A008543 a(n) = (-1)^n*Sum_{k=0..n} 6^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 A008543 G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(6*k-1)/(1 - x*(6*k+6)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013
%F A008543 a(n) = 6^n * Gamma(n+5/6) / Gamma(5/6). - _Vaclav Kotesovec_, Jan 28 2015
%F A008543 D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F A008543 From _Nikolaos Pantelidis_, May 22 2022: (Start)
%F A008543 G.f.: 1/G(0), where G(k) = 1 - (12*k+5)*x - 6*(k+1)*(6*k+5)*x^2/G(k+1) (a continued fraction);
%F A008543 which starts 1/(1-5*x-30*x^2/(1-17*x-132*x^2/(1-29*x-306*x^2/(1-41*x-552*x^2/(1-53*x-870*x^2/(1-65*x-1260*x^2/(1-...))))))) (a Jacobi continued fraction).
%F A008543 (End)
%F A008543 Sum_{n>=0} 1/a(n) = 1 + (e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - _Amiram Eldar_, Dec 18 2022
%p A008543 f := n->product( (6*k-1),k=0..n);
%t A008543 FoldList[Times,1,6Range[0,15]+5]  (* _Harvey P. Dale_, Feb 20 2011 *)
%t A008543 Table[6^n*Pochhammer[5/6, n], {n, 0, 20}] (* _G. C. Greubel_, Dec 03 2019 *)
%t A008543 CoefficientList[Series[(1 - 6x)^(-5/6), {x, 0, 20}], x] Range[0, 20]! (* _Nikolaos Pantelidis_, Jan 31 2023 *)
%o A008543 (PARI) a(n)=prod(k=1,n,6*k-1) \\ _Charles R Greathouse IV_, Aug 17 2011
%o A008543 (Magma) [Round(6^n*Gamma(n+5/6)/Gamma(5/6)): n in [0..20]]; // _G. C. Greubel_, Dec 03 2019
%o A008543 (Sage) [6^n*rising_factorial(5/6, n) for n in (0..20)] # _G. C. Greubel_, Dec 03 2019
%Y A008543 a(n) = A013988(n+1, 1) (first column of triangle).
%Y A008543 Cf. A004994, A034787, A049308, A047058, A048994, A051151.
%K A008543 nonn,easy
%O A008543 0,2
%A A008543 Joe Keane (jgk(AT)jgk.org)