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A229267 Number of lattice paths from {n}^n to {0}^n using steps that decrement one component or all components by 1.

%I A229267 #28 Dec 27 2013 03:54:00
%S A229267 1,1,13,2371,67982041,629799991355641,2672932604015450235911761,
%T A229267 7364217994146042440421602767480184881,
%U A229267 18165821273625565354157327818616137066973745155992321,53130704578476340997304138835621075610747224340706918846011664495415681
%N A229267 Number of lattice paths from {n}^n to {0}^n using steps that decrement one component or all components by 1.
%H A229267 Alois P. Heinz, <a href="/A229267/b229267.txt">Table of n, a(n) for n = 0..25</a>
%F A229267 a(n) = Sum_{j=0..n} multinomial(n+(n-1)*j; n-j, {j}^n) for n>1, a(0) = a(1) = 1.
%e A229267 a(2) = 3*3 + 2*2 = 13:
%e A229267 .           (0,2)
%e A229267 .          /     \
%e A229267 .     (1,2)-------(0,1)
%e A229267 .    /     \     /     \
%e A229267 (2,2)-------(1,1)-------(0,0)
%e A229267 .    \     /     \     /
%e A229267 .     (2,1)-------(1,0)
%e A229267 .          \     /
%e A229267 .           (2,0)
%p A229267 with(combinat):
%p A229267 a:= n-> `if`(n<2, 1, add(multinomial(n+(n-1)*j, n-j, j$n), j=0..n)):
%p A229267 seq(a(n), n=0..10);
%t A229267 multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := If[n < 2, 1, Sum[multinomial[n+(n-1)*j, Join[{n-j}, Array[j&, n]]], {j, 0, n}]]; Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A229267 Main diagonal of A229142.
%K A229267 nonn
%O A229267 0,3
%A A229267 _Alois P. Heinz_, Sep 23 2013