A229676 - OEIS

%I #13 Oct 07 2018 18:29:42

%S 1,362881,12504639772801,1080492192338314694401,

%T 140810184334251776225321193601,

%U 23183593018924832394604719137184142081,4439414110286267003192333763481728593177802241,944848564471993704169724618186222285154304912036663681

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

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

%H Alois P. Heinz, <a href="/A229676/b229676.txt">Table of n, a(n) for n = 0..100</a>

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

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

%F 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

%p with(combinat):

%p a:= n-> add(multinomial(n+8*k, n-k, k$9), k=0..n):

%p seq(a(n), n=0..10);

%t 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 *)

%Y Column k = 9 of A229142.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Sep 27 2013