A004996 a(n) = 6^n/n! * Product_{k=0..n-1} (6*k - 1).

1, -6, -90, -1980, -50490, -1393524, -40412196, -1212365880, -37280250810, -1168114525380, -37146041907084, -1195427166827976, -38851382921909220, -1273129932671794440, -42013287778169216520

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A004993.

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

seq(6^n*product(6*k-1, k = 0..n-1)/n!, n = 0..20); # G. C. Greubel, Aug 20 2019

Mathematica

Table[6^(2*n)*Pochhammer[-1/6, n]/n!, {n,0,20}] (* G. C. Greubel, Aug 20 2019 *)

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) ~ -1/6*Gamma(5/6)^-1*n^(-7/6)*6^(2*n)*{1 + 7/72*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

a(n) = 6^(2*n) * binomial(n-7/6, n). - G. C. Greubel, Aug 20 2019

D-finite with recurrence: n*a(n) +6*(-6*n+7)*a(n-1)=0. - R. J. Mathar, Jan 17 2020