A229677 a(n) = Sum_{k = 0..n} Product_{j = 0..9} C(n+j*k,k).

1, 3628801, 2375880907276801, 4386797386179342934060801, 12868640117405297821759744777996801, 49120459033702373637913562847507823210617601, 222254155614179529476178258638452174287098861960755201, 1132660294172702489573582429384603543633942385302181948349459201

Offset: 0

Author

Alois P. Heinz, Sep 27 2013

Keywords

• nonn
• easy

Comments

Number of lattice paths from {n}^10 to {0}^10 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 = 10 of A229142.

Programs

Maple

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

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := Sum[multinomial[n + 9*k, Join[{n - k}, Array[k&, 10]]], {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+9*k; n-k, {k}^10).

G.f.: Sum_{k >= 0} (10*k)!/k!^10 * x^k / (1-x)^(10*k+1).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 1814401*x^2 + 791960304240001*x^3 + 1096699347338442061435201*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016