A004381 Binomial coefficient C(8n,n).

1, 8, 120, 2024, 35960, 658008, 12271512, 231917400, 4426165368, 85113005120, 1646492110120, 32006008361808, 624668654531480, 12233149001721760, 240260199935164200, 4730523156632595024, 93343021201262177400, 1845382436487682488000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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].

Crossrefs

Row 8 of A060539.

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).

Programs

Magma

[Binomial(8*n, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Mathematica

Table[Binomial[8 n, n], {n, 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

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)

From Seiichi Manyama, Aug 16 2025: (Start)

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*n+1,k).

G.f.: 1/(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.

G.f.: g/(8-7*g) where g = 1+x*g^8 is the g.f. of A007556. (End)