A229675 a(n) = Sum_{k = 0..n} Product_{j = 0..7} C(n+j*k,k).

1, 40321, 81730010881, 369400348294790401, 2390471064720364776796801, 18975660656355118819906214670721, 171890067585060168829713844899790066561, 1707759022485971054271963683059722310362986881, 18165821273625565354157327818616137066973745155992321

Offset: 0

Author

Alois P. Heinz, Sep 27 2013

Keywords

• nonn
• easy

Comments

Number of lattice paths from {n}^8 to {0}^8 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 = 8 of A229142.

Programs

Maple

with(combinat):
a:= n-> add(multinomial(n+7*k, n-k, k$8), k=0..n):
seq(a(n), n=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n + 7*k, Join[{n - k}, Array[k&, 8]]], {k, 0, n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
Table[Sum[Product[Binomial[n+j*k,k],{j,0,7}],{k,0,n}],{n,0,10}] (* Harvey P. Dale, Aug 25 2014 *)

Formula

a(n) = Sum_{k = 0..n} multinomial(n+7*k; n-k, {k}^8).

G.f.: Sum_{k >= 0} (8*k)!/k!^8 * x^k / (1-x)^(8*k+1).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 20161*x^2 + 27243357121*x^3 + 92350114520267521*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016