A004381 Binomial coefficient C(8n,n).

1, 8, 120, 2024, 35960, 658008, 12271512, 231917400, 4426165368, 85113005120, 1646492110120, 32006008361808, 624668654531480, 12233149001721760, 240260199935164200, 4730523156632595024

Offset

0,2

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Links

T. D. Noe, Table of n, a(n) for n=0..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

a(n) = C(8*n-1,n-1)*C(64*n^2,2)/(3*n*C(8*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014

From Ilya Gutkovskiy, Jan 16 2017: (Start)

O.g.f.: 7F6(1/8,1/4,3/8,1/2,5/8,3/4,7/8; 1/7,2/7,3/7,4/7,5/7,6/7; 16777216*x/823543).

E.g.f.: 7F7(1/8,1/4,3/8,1/2,5/8,3/4,7/8; 1/7,2/7,3/7,4/7,5/7,6/7,1; 16777216*x/823543).

a(n) ~ 2^(24*n+1)/(sqrt(Pi*n)*7^(7*n+1/2)). (End)

From Peter Bala, Feb 20 2022: (Start)

The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 7*A(x))^7 + (8^8)*x*A(x)^8 = 0.

Sum_{n >= 1} a(n)*( x*(7*x + 8)^7/(8^8*(1 + x)^8) )^n = x. (End)

Mathematica

Table[Binomial[8 n, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

Prog

(Magma) [Binomial(8*n, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014
(Python)
from math import comb
def A004381(n): return comb(n<<3, n) # Chai Wah Wu, Aug 01 2023

Crossrefs

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A169958 - A169961 (k = 9 thru 12).

Sequence in context: A249641 A045899 A165231 * A166179 A130979 A239226

Adjacent sequences: A004378 A004379 A004380 * A004382 A004383 A004384

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved