A292898 Array read by ascending antidiagonals, A(m, n) = Sum_{k=1..m}(-1)^(k-n-m)* hypergeom([k, k-n-m], [], 1) for m>=1 and n>=0.
1, 1, 0, 3, 2, 1, 8, 7, 5, 2, 31, 30, 27, 20, 9, 147, 146, 142, 129, 97, 44, 853, 852, 847, 826, 755, 574, 265, 5824, 5823, 5817, 5786, 5652, 5187, 3973, 1854, 45741, 45740, 45733, 45690, 45463, 44462, 40923, 31520, 14833
Offset: 0
Examples
Array starts: [m\n] 0 1 2 3 4 5 6 ------------------------------------------------------------------- [1] 1, 0, 1, 2, 9, 44, 265, ... [A000166] [2] 1, 2, 5, 20, 97, 574, 3973, ... [A259834(n+2)] [3] 3, 7, 27, 129, 755, 5187, 40923, ... [A292897] [4] 8, 30, 142, 826, 5652, 44462, 394970, ... [5] 31, 146, 847, 5786, 45463, 403514, 3990679, ... [6] 147, 852, 5817, 45690, 405423, 4008768, 43692933, ... [7] 853, 5823, 45733, 405779, 4012101, 43727687, 520723477, ... A003470,A193464,A293295. Displayed as a triangle: [1] 1; [2] 1, 0; [3] 3, 2, 1; [4] 8, 7, 5, 2; [5] 31, 30, 27, 20, 9; [6] 147, 146, 142, 129, 97, 44; [7] 853, 852, 847, 826, 755, 574, 265; [8] 5824, 5823, 5817, 5786, 5652, 5187, 3973, 1854; A003470,A193464,A293295. This triangle has row sums A193463.
Programs
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Maple
A := (m, n) -> add((-1)^(k-n-m)*hypergeom([k, k-n-m], [], 1), k=1..m): seq(lprint(seq(simplify(A(m, n)), n=0..6)), m=1..7);
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Mathematica
A[m_, n_] := Sum[(-1)^(k-n-m) HypergeometricPFQ[{k, k-n-m},{}, 1], {k, 1, m} ]; Table[Table[A[m, n], {n,0,6}], {m,1,7}]