A229676 a(n) = Sum_{k = 0..n} Product_{j = 0..8} C(n+j*k,k).

1, 362881, 12504639772801, 1080492192338314694401, 140810184334251776225321193601, 23183593018924832394604719137184142081, 4439414110286267003192333763481728593177802241, 944848564471993704169724618186222285154304912036663681

Offset: 0

Author

Alois P. Heinz, Sep 27 2013

Keywords

• nonn
• easy

Comments

Number of lattice paths from {n}^9 to {0}^9 using steps that decrement one component or all components by 1.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Column k = 9 of A229142.

Programs

Maple

with(combinat):
a:= n-> add(multinomial(n+8*k, n-k, k$9), k=0..n):
seq(a(n), n=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n + 8*k, Join[{n - k}, Array[k&, 9]]], {k, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Formula

a(n) = Sum_{k = 0..n} multinomial(n+8*k; n-k, {k}^9).

G.f.: Sum_{k >= 0} (9*k)!/k!^9 * x^k / (1-x)^(9*k+1).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 181441*x^2 + 4168213439041*x^3 + 270123052269252349441*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016