%I #35 Sep 08 2022 08:44:33
%S 1,6,126,3276,93366,2800980,86830380,2753763480,88808872230,
%T 2901089826180,95735964263940,3185396629145640,106710787076378940,
%U 3595332672265690440,121727691903852662040,4138741524730990509360
%N a(n) = (6^n/n!)*Product_{k=0..n-1} (6*k + 1).
%H Seiichi Manyama, <a href="/A004993/b004993.txt">Table of n, a(n) for n = 0..500</a>
%H A. Straub, V. H. Moll, T. Amdeberhan, <a href="http://dx.doi.org/10.4064/aa140-1-2">The p-adic valuation of k-central binomial coefficients</a>, Acta Arith. 140 (1) (2009) 31-41, eq (1.10)
%H <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F G.f.: (1 - 36*x)^(-1/6).
%F a(n) ~ Gamma(1/6)^-1*n^(-5/6)*6^(2*n)*{1 - 5/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F a(n) = (-36)^n*binomial(-1/6, n). - _Peter Luschny_, Oct 23 2018
%F D-finite with recurrence: n*a(n) +6*(-6*n+5)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%p A004993 := n -> (-36)^n*binomial(-1/6, n):
%p seq(A004993(n), n=0..16); # _Peter Luschny_, Oct 23 2018
%t Table[(-36)^n Binomial[-1/6, n], {n, 0, 15}] (* _Jean-François Alcover_, Jun 02 2019, after _Peter Luschny_ *)
%o (PARI) vector(20, n, n--; 6^n*prod(j=0,n-1, 6*j+1)/n! ) \\ _G. C. Greubel_, Aug 20 2019
%o (Magma) [1] cat [6^n*(&*[6*k+1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // _G. C. Greubel_, Aug 20 2019
%o (Sage) [6^(2*n)*rising_factorial(1/6, n)/factorial(n) for n in (0..20)] # _G. C. Greubel_, Aug 20 2019
%o (GAP) List([0..20], n-> 6^n*Product([0..n-1], k-> 6*k+1)/Factorial(n) ); # _G. C. Greubel_, Aug 20 2019
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
%E Corrected by _Franklin T. Adams-Watters_, Oct 25 2006