A004994 - OEIS

%I #35 Sep 08 2022 08:44:33

%S 1,30,990,33660,1161270,40412196,1414426860,49707001080,1752171788070,

%T 61910069845140,2191616472517956,77702765843818440,

%U 2758448187455554620,98031004815728171880,3487102885588044971160

%N a(n) = (6^n/n!)*Product_{k=0..n-1} (6*k + 5).

%H G. C. Greubel, <a href="/A004994/b004994.txt">Table of n, a(n) for n = 0..635</a>

%F G.f.: (1 - 36*x)^(-5/6).

%F a(n) ~ Gamma(5/6)^-1*n^(-1/6)*6^(2*n)*(1 - (5/72)*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

%F a(n) = Product_{k=1..n} (36 - 6/k). - _Michel Lagneau_, Sep 16 2012

%F a(n) = (-36)^n*binomial(-5/6, n). - _Peter Luschny_, Oct 23 2018

%F E.g.f.: L_{-5/6}(36*x) where L_{k}(x) is the Laguerre polynomial. - _Stefano Spezia_, Aug 21 2019

%F D-finite with recurrence: n*a(n) +6*(-6*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020

%p A004994 := n -> (-36)^n*binomial(-5/6, n):

%p seq(A004994(n), n=0..16); # _Peter Luschny_, Oct 23 2018

%t Table[Product[36-6/k,{k,n}],{n,0,20}] (* or *) FoldList[Times,1,36-6/Range[20]] (* _Harvey P. Dale_, Feb 27 2013 *)

%o (PARI) vector(20, n, n--; 6^n*prod(j=0,n-1, 6*j+5)/n! ) \\ _G. C. Greubel_, Aug 20 2019

%o (Magma) [1] cat [6^n*(&*[6*k+5: k in [0..n-1]])/Factorial(n): n in [1..20]]; // _G. C. Greubel_, Aug 20 2019

%o (Sage) [6^(2*n)*rising_factorial(5/6, 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+5)/Factorial(n) ); # _G. C. Greubel_, Aug 20 2019

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)