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%I #46 Jul 10 2020 22:09:51
%S 1,1,1,1,1,1,1,3,1,1,1,7,13,1,1,1,25,115,63,1,1,1,121,2641,2371,321,1,
%T 1,1,721,114121,392641,54091,1683,1,1,1,5041,7489441,169417921,
%U 67982041,1307377,8989,1,1,1,40321,681120721,137322405361,308238414121,12838867105,32803219,48639,1,1
%N Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j - Product_{j=1..k} x_j) for k>1. - _Seiichi Manyama_, Jul 10 2020
%H Alois P. Heinz, <a href="/A229142/b229142.txt">Antidiagonals n = 0..44, flattened</a>
%F A(n,k) = Sum_{j=0..n} multinomial(n+(k-1)*j; n-j, {j}^k) for k>1, A(n,0) = A(n,1) = 1.
%F G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1-x)^(k*j+1). for k>1. - _Seiichi Manyama_, Jul 10 2020
%e A(1,3) = 3*2+1 = 7:
%e (0,1,1)-(0,0,1)
%e / X \
%e (1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
%e \ \ X / /
%e \ (1,1,0)-(1,0,0) /
%e `---------------´
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 7, 25, 121, ...
%e 1, 1, 13, 115, 2641, 114121, ...
%e 1, 1, 63, 2371, 392641, 169417921, ...
%e 1, 1, 321, 54091, 67982041, 308238414121, ...
%e 1, 1, 1683, 1307377, 12838867105, 629799991355641, ...
%p with(combinat):
%p A:= (n,k)-> `if`(k<2, 1, add(multinomial(n+(k-1)*j, n-j, j$k), j=0..n)):
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%t a[_, 0] = a[_, 1] = 1; a[n_, k_] := Sum[Product[Binomial[n+j*m, m], {j, 0, k-1}], {m, 0, n}]; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 11 2013 *)
%Y Columns k=0+1, 2-10 give: A000012, A001850, A081798, A082488, A082489, A229049, A229674, A229675, A229676, A229677.
%Y Rows n=0-1 give: A000012, A038507 (for k>1).
%Y Main diagonal gives: A229267.
%Y Cf. A060854, A227578, A227655, A225094, A210472, A262809, A263159.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 23 2013