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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. 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified June 15 19:43 EDT 2024. Contains 373410 sequences. (Running on oeis4.)