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A008978 a(n) = (5*n)!/(n!)^5. 9
1, 120, 113400, 168168000, 305540235000, 623360743125120, 1370874167589326400, 3177459078523411968000, 7656714453153197981835000, 19010638202652030712978200000, 48334775757901219912115629238400 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

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

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)

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

CROSSREFS

Cf. A000984, A006480, A008977,  A002894, A002897, A186420, A188662.

Sequence in context: A184887 A159735 A157879 * A077692 A184127 A068296

Adjacent sequences:  A008975 A008976 A008977 * A008979 A008980 A008981

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 2 12:56 EST 2016. Contains 278678 sequences.