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

%I A229674 #18 Jan 05 2018 17:45:51
%S A229674 1,5041,681120721,182519583427441,66479594535132382801,
%T A229674 28839847393654717358640241,14008219182384095498700747103921,
%U A229674 7364217994146042440421602767480184881,4104535666620019954310735707010727096941521,2392974957158188560683670847243199936518562761441
%N A229674 a(n) = Sum_{k = 0..n} Product_{j = 0..6} C(n+j*k,k).
%C A229674 Number of lattice paths from {n}^7 to {0}^7 using steps that decrement one component or all components by 1.
%H A229674 Alois P. Heinz, <a href="/A229674/b229674.txt">Table of n, a(n) for n = 0..100</a>
%F A229674 a(n) = Sum_{k = 0..n} multinomial(n+6*k; n-k, {k}^7).
%F A229674 G.f.: Sum_{k >= 0} (7*k)!/k!^7 * x^k / (1-x)^(7*k+1).
%F A229674 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
%p A229674 with(combinat):
%p A229674 a:= n-> add(multinomial(n+6*k, n-k, k$7), k=0..n):
%p A229674 seq(a(n), n=0..10);
%t A229674 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 *)
%Y A229674 Column k = 7 of A229142.
%K A229674 nonn,easy
%O A229674 0,2
%A A229674 _Alois P. Heinz_, Sep 27 2013