A004994 a(n) = (6^n/n!)*Product_{k=0..n-1} (6*k + 5).

1, 30, 990, 33660, 1161270, 40412196, 1414426860, 49707001080, 1752171788070, 61910069845140, 2191616472517956, 77702765843818440, 2758448187455554620, 98031004815728171880, 3487102885588044971160

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..635

Programs

GAP

List([0..20], n-> 6^n*Product([0..n-1], k-> 6*k+5)/Factorial(n) ); # G. C. Greubel, Aug 20 2019

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

Maple

A004994 := n -> (-36)^n*binomial(-5/6, n):
seq(A004994(n), n=0..16); # Peter Luschny, Oct 23 2018

Mathematica

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

PARI

vector(20, n, n--; 6^n*prod(j=0,n-1, 6*j+5)/n! ) \\ G. C. Greubel, Aug 20 2019

Sage

[6^(2*n)*rising_factorial(5/6, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

Formula

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

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

a(n) = Product_{k=1..n} (36 - 6/k). - Michel Lagneau, Sep 16 2012

a(n) = (-36)^n*binomial(-5/6, n). - Peter Luschny, Oct 23 2018

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

D-finite with recurrence: n*a(n) +6*(-6*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020