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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.

%I A292898 #14 Jan 06 2018 18:39:54
%S A292898 1,1,0,3,2,1,8,7,5,2,31,30,27,20,9,147,146,142,129,97,44,853,852,847,
%T A292898 826,755,574,265,5824,5823,5817,5786,5652,5187,3973,1854,45741,45740,
%U A292898 45733,45690,45463,44462,40923,31520,14833
%N 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.
%e A292898 Array starts:
%e A292898 [m\n]   0       1      2       3        4         5          6
%e A292898 -------------------------------------------------------------------
%e A292898 [1]     1,      0,     1,      2,       9,       44,       265, ...  [A000166]
%e A292898 [2]     1,      2,     5,     20,      97,      574,      3973, ...  [A259834(n+2)]
%e A292898 [3]     3,      7,    27,    129,     755,     5187,     40923, ...  [A292897]
%e A292898 [4]     8,     30,   142,    826,    5652,    44462,    394970, ...
%e A292898 [5]    31,    146,   847,   5786,   45463,   403514,   3990679, ...
%e A292898 [6]   147,    852,  5817,  45690,  405423,  4008768,  43692933, ...
%e A292898 [7]   853,   5823, 45733, 405779, 4012101, 43727687, 520723477, ...
%e A292898   A003470,A193464,A293295.
%e A292898 Displayed as a triangle:
%e A292898 [1]     1;
%e A292898 [2]     1,      0;
%e A292898 [3]     3,      2,     1;
%e A292898 [4]     8,      7,     5,    2;
%e A292898 [5]    31,     30,    27,   20,    9;
%e A292898 [6]   147,    146,   142,  129,   97,   44;
%e A292898 [7]   853,    852,   847,  826,  755,  574,  265;
%e A292898 [8]  5824,   5823,  5817, 5786, 5652, 5187, 3973, 1854;
%e A292898   A003470,A193464,A293295.
%e A292898 This triangle has row sums A193463.
%p A292898 A := (m, n) -> add((-1)^(k-n-m)*hypergeom([k, k-n-m], [], 1), k=1..m):
%p A292898 seq(lprint(seq(simplify(A(m, n)), n=0..6)), m=1..7);
%t A292898 A[m_, n_] :=  Sum[(-1)^(k-n-m) HypergeometricPFQ[{k, k-n-m},{}, 1], {k, 1, m} ];
%t A292898 Table[Table[A[m, n], {n,0,6}], {m,1,7}]
%Y A292898 Cf. A000166, A003470, A193463, A193464, A259834, A292897, A293295.
%K A292898 nonn,tabl
%O A292898 0,4
%A A292898 _Peter Luschny_, Oct 05 2017