OFFSET
0,2
COMMENTS
Number of paths of length 5n in Z^5 from (0,0,0,0,0) to (n,n,n,n,n).
Entry # 1 in the "Big Table" of Almkvist et al. - Peter Bala, Sep 17 2025
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
G. Almkvist et al., Tables of Calabi-Yau Equations, arXiv:math/0507430 [math.AG], 2005-2010, p. 10.
V. Batyrev, Review of "Mirror Symmetry and Algebraic Geometry", by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.
R. M. Dickau, 5-D shortest path diagrams
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
FORMULA
a(n) ~ 5^(5*n+1/2) / (4 * Pi^2 * n^2). - Vaclav Kotesovec, Mar 07 2014
From Peter Bala, Jul 12 2016: (Start)
a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n)*binomial(5*n,n) = ( [x^n](1 + x)^(2*n) ) * ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) * ( [x^n](1 + x)^(5*n) ) = [x^n]( F(x)^(120*n) ), where F(x) = 1 + x + 353*x^2 + 318986*x^3 + 408941594*x^4 + 633438203535*x^5 + 1105336091531052*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977, A186420 and A188662. (End)
From Peter Bala, Jul 17 2016: (Start)
a(n) = Sum_{k = 0..4*n} (-1)^k*binomial(5*n,n + k)*binomial(n + k,k)^5.
a(n) = Sum_{k = 0..5*n} (-1)^(n+k)*binomial(5*n,k)*binomial(n + k,k)^5. (End)
From Ilya Gutkovskiy, Nov 23 2017: (Start)
O.g.f.: 4F3(1/5,2/5,3/5,4/5; 1,1,1; 3125*x).
E.g.f.: 4F4(1/5,2/5,3/5,4/5; 1,1,1,1; 3125*x). (End)
From Peter Bala, Feb 16 2020: (Start)
The supercongruence a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) holds for all primes p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
a(n) = [(x*y*z*u)^n] (1 + x + y + z + u )^(5*n). (End)
a(n) = 120*A322252(n). - R. J. Mathar, Jun 21 2023
a(n) = a(n-1)*5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/n^4. - Neven Sajko, Jul 21 2023
From Peter Bala, Sep 19 2025: (Start)
The series E(x) = exp( Sum_{n >= 1} (1/120)*a(n)*x^n/n ) = 1 + x + 473*x^2 + 467606*x^3 + 637121154*x^4 + 1039792179805*x^5 + 1905441263652576*x^6 + ... appears to have integer coefficients.
For positive integer r, define sequences {u(n) : n >= 0} and {v(n) : n >= 0} by setting u(n) = [x^n] E(x)^(r*n) and v(n) = [x^n] F(x)^(r*n), where F(x) is defined above in the posting dated Jul 12 2016. We conjecture that the supercongruences u(m*p^k) == u(m*p^(k-1)) ( mod p^(3*k) ) and v(m*p^k) == v(m*p^(k-1)) ( mod p^(3*k) ) hold for all primes p >= 5 and positive integers m and k. (End)
MATHEMATICA
Table[(5 n)!/(n)!^5, {n, 0, 20}] (* Vincenzo Librandi, Mar 08 2014 *)
PROG
(Magma) [Factorial(5*n)/Factorial(n)^5: n in [0..10]]; // Vincenzo Librandi, Mar 08 2014
(PARI) a(n) = (5*n)!/(n!)^5; \\ Michel Marcus, Mar 08 2014
(Python)
from math import prod, comb
def A008978(n): return prod(comb(i*n, n) for i in range(2, 6)) # Chai Wah Wu, Feb 17 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved