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A316674 Number A(n,k) of paths from {0}^k to {n}^k that always move closer to {n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A316674 #27 Nov 04 2021 06:01:56
%S A316674 1,1,1,1,1,1,1,3,2,1,1,13,26,4,1,1,75,818,252,8,1,1,541,47834,64324,
%T A316674 2568,16,1,1,4683,4488722,42725052,5592968,26928,32,1,1,47293,
%U A316674 617364026,58555826884,44418808968,515092048,287648,64,1
%N A316674 Number A(n,k) of paths from {0}^k to {n}^k that always move closer to {n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A316674 A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n. - _Andrew Howroyd_, Jan 23 2020
%H A316674 Alois P. Heinz, <a href="/A316674/b316674.txt">Antidiagonals n = 0..48, flattened</a>
%F A316674 A(n,k) = A262809(n,k) * A011782(n) for k>0, A(n,0) = 1.
%F A316674 A(n,k) = Sum_{j=0..n*k} binomial(j+n-1,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - _Andrew Howroyd_, Jan 23 2020
%e A316674 Square array A(n,k) begins:
%e A316674   1,  1,     1,         1,              1,                    1, ...
%e A316674   1,  1,     3,        13,             75,                  541, ...
%e A316674   1,  2,    26,       818,          47834,              4488722, ...
%e A316674   1,  4,   252,     64324,       42725052,          58555826884, ...
%e A316674   1,  8,  2568,   5592968,    44418808968,      936239675880968, ...
%e A316674   1, 16, 26928, 515092048, 50363651248560, 16811849850663255376, ...
%p A316674 A:= (n, k)-> `if`(k=0, 1, ceil(2^(n-1))*add(add((-1)^i*
%p A316674      binomial(j, i)*binomial(j-i, n)^k, i=0..j), j=0..k*n)):
%p A316674 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A316674 A[n_, k_] := Sum[If[k == 0, 1, Binomial[j+n-1, n]^k] Sum[(-1)^(i-j)* Binomial[i, j], {i, j, n k}], {j, 0, n k}];
%t A316674 Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Nov 04 2021 *)
%o A316674 (PARI) T(n,k)={my(m=n*k); sum(j=0, m, binomial(j+n-1,n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ _Andrew Howroyd_, Jan 23 2020
%Y A316674 Columns k=0..3 give: A000012, A011782, A052141, A316673.
%Y A316674 Rows n=0..2 give: A000012, A000670, A059516.
%Y A316674 Main diagonal gives A316677.
%Y A316674 Cf. A011782, A219727, A262809, A331315, A331485, A331636.
%K A316674 nonn,tabl,walk
%O A316674 0,8
%A A316674 _Alois P. Heinz_, Jul 10 2018