A229674 a(n) = Sum_{k = 0..n} Product_{j = 0..6} C(n+j*k,k).

1, 5041, 681120721, 182519583427441, 66479594535132382801, 28839847393654717358640241, 14008219182384095498700747103921, 7364217994146042440421602767480184881, 4104535666620019954310735707010727096941521, 2392974957158188560683670847243199936518562761441

Offset: 0

Author

Alois P. Heinz, Sep 27 2013

Keywords

• nonn
• easy

Comments

Number of lattice paths from {n}^7 to {0}^7 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 = 7 of A229142.

Programs

Maple

with(combinat):
a:= n-> add(multinomial(n+6*k, n-k, k$7), k=0..n):
seq(a(n), n=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n+6*k, Join[{n-k}, Array[k&, 7]]], {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+6*k; n-k, {k}^7).

G.f.: Sum_{k >= 0} (7*k)!/k!^7 * x^k / (1-x)^(7*k+1).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2521*x^2 + 227042761*x^3 + 45630126074821*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016