A004993 a(n) = (6^n/n!)*Product_{k=0..n-1} (6*k + 1).

1, 6, 126, 3276, 93366, 2800980, 86830380, 2753763480, 88808872230, 2901089826180, 95735964263940, 3185396629145640, 106710787076378940, 3595332672265690440, 121727691903852662040, 4138741524730990509360

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• A. Straub, V. H. Moll, T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10)
• Index entries for sequences related to the Josephus Problem

Programs

GAP

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

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

Maple

A004993 := n -> (-36)^n*binomial(-1/6, n):
seq(A004993(n), n=0..16); # Peter Luschny, Oct 23 2018

Mathematica

Table[(-36)^n Binomial[-1/6, n], {n, 0, 15}] (* Jean-François Alcover, Jun 02 2019, after Peter Luschny *)

PARI

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

Sage

[6^(2*n)*rising_factorial(1/6, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

Formula

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

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

a(n) = (-36)^n*binomial(-1/6, n). - Peter Luschny, Oct 23 2018

D-finite with recurrence: n*a(n) +6*(-6*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

Extensions

Corrected by Franklin T. Adams-Watters, Oct 25 2006