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.

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, 2568, 16, 1, 1, 4683, 4488722, 42725052, 5592968, 26928, 32, 1, 1, 47293, 617364026, 58555826884, 44418808968, 515092048, 287648, 64, 1

Offset

0,8

Comments

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

Links

Alois P. Heinz, Antidiagonals n = 0..48, flattened

Formula

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

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

Example

Square array A(n,k) begins:
  1,  1,     1,         1,              1,                    1, ...
  1,  1,     3,        13,             75,                  541, ...
  1,  2,    26,       818,          47834,              4488722, ...
  1,  4,   252,     64324,       42725052,          58555826884, ...
  1,  8,  2568,   5592968,    44418808968,      936239675880968, ...
  1, 16, 26928, 515092048, 50363651248560, 16811849850663255376, ...

Maple

A:= (n, k)-> `if`(k=0, 1, ceil(2^(n-1))*add(add((-1)^i*
     binomial(j, i)*binomial(j-i, n)^k, i=0..j), j=0..k*n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

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}];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 04 2021 *)

Prog

(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

Crossrefs

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

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

Main diagonal gives A316677.

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

Sequence in context: A316564 A214742 A204124 * A101479 A136170 A245188

Adjacent sequences: A316671 A316672 A316673 * A316675 A316676 A316677

Keyword

nonn,tabl,walk

Author

Alois P. Heinz, Jul 10 2018

Status

approved