cp's OEIS Frontend

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

%I A229677 #13 Oct 07 2018 18:26:54
%S A229677 1,3628801,2375880907276801,4386797386179342934060801,
%T A229677 12868640117405297821759744777996801,
%U A229677 49120459033702373637913562847507823210617601,222254155614179529476178258638452174287098861960755201,1132660294172702489573582429384603543633942385302181948349459201
%N A229677 a(n) = Sum_{k = 0..n} Product_{j = 0..9} C(n+j*k,k).
%C A229677 Number of lattice paths from {n}^10 to {0}^10 using steps that decrement one component or all components by 1.
%H A229677 Alois P. Heinz, <a href="/A229677/b229677.txt">Table of n, a(n) for n = 0..100</a>
%F A229677 a(n) = Sum_{k = 0..n} multinomial(n+9*k; n-k, {k}^10).
%F A229677 G.f.: Sum_{k >= 0} (10*k)!/k!^10 * x^k / (1-x)^(10*k+1).
%F A229677 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
%p A229677 with(combinat):
%p A229677 a:= n-> add(multinomial(n+9*k, n-k, k$10), k=0..n):
%p A229677 seq(a(n), n=0..10);
%t A229677 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 *)
%Y A229677 Column k = 10 of A229142.
%K A229677 nonn,easy
%O A229677 0,2
%A A229677 _Alois P. Heinz_, Sep 27 2013