A331636 -id:A331636

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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.

+10
11

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

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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.

KEYWORD

nonn,tabl,walk

AUTHOR

Alois P. Heinz, Jul 10 2018

STATUS

approved

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