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)
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)
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
(PARI) a(n) = binomial(8*n, n); \\ Amiram Eldar, Sep 09 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved