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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.
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%I #27 Nov 04 2021 06:01:56

%S 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 2568,16,1,1,4683,4488722,42725052,5592968,26928,32,1,1,47293,

%U 617364026,58555826884,44418808968,515092048,287648,64,1

%N 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 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 Alois P. Heinz, <a href="/A316674/b316674.txt">Antidiagonals n = 0..48, flattened</a>

%F A(n,k) = A262809(n,k) * A011782(n) for k>0, A(n,0) = 1.

%F 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 Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 3, 13, 75, 541, ...

%e 1, 2, 26, 818, 47834, 4488722, ...

%e 1, 4, 252, 64324, 42725052, 58555826884, ...

%e 1, 8, 2568, 5592968, 44418808968, 936239675880968, ...

%e 1, 16, 26928, 515092048, 50363651248560, 16811849850663255376, ...

%p A:= (n, k)-> `if`(k=0, 1, ceil(2^(n-1))*add(add((-1)^i*

%p binomial(j, i)*binomial(j-i, n)^k, i=0..j), j=0..k*n)):

%p seq(seq(A(n, d-n), n=0..d), d=0..10);

%t 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 Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Nov 04 2021 *)

%o (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 Columns k=0..3 give: A000012, A011782, A052141, A316673.

%Y Rows n=0..2 give: A000012, A000670, A059516.

%Y Main diagonal gives A316677.

%Y Cf. A011782, A219727, A262809, A331315, A331485, A331636.

%K nonn,tabl,walk

%O 0,8

%A _Alois P. Heinz_, Jul 10 2018