# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a316674 Showing 1-1 of 1 %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, Antidiagonals n = 0..48, flattened %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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE