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%I #22 Mar 20 2020 04:54:08
%S 0,0,1,0,1,0,0,1,1,1,0,1,2,3,2,0,1,3,7,11,9,0,1,4,13,32,53,44,0,1,5,
%T 21,71,181,309,265,0,1,6,31,134,465,1214,2119,1854,0,1,7,43,227,1001,
%U 3539,9403,16687,14833,0,1,8,57,356,1909,8544,30637,82508,148329,133496
%N Square array read by antidiagonals: A(k, n) = (-1)^(n+1)* hypergeom([k, -n+1], [], 1) for n>0 and A(k,0) = 0 (n>=0, k>=1).
%e k\n
%e [1], 0, 1, 0, 1, 2, 9, 44, 265, 1854, ... A000166
%e [2], 0, 1, 1, 3, 11, 53, 309, 2119, 16687, ... A000255
%e [3], 0, 1, 2, 7, 32, 181, 1214, 9403, 82508, ... A000153
%e [4], 0, 1, 3, 13, 71, 465, 3539, 30637, 296967, ... A000261
%e [5], 0, 1, 4, 21, 134, 1001, 8544, 81901, 870274, ... A001909
%e [6], 0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319, ... A001910
%e [7], 0, 1, 6, 43, 356, 3333, 34754, 398959, 4996032, ... A176732
%e [8], 0, 1, 7, 57, 527, 5441, 61959, 770713, 10391023, ... A176733
%e The referenced sequences may have a different offset or other small deviations.
%p A := (k,n) -> `if`(n<2,n,hypergeom([k,-n+1],[],1)*(-1)^(n+1));
%p seq(print(seq(round(evalf(A(k,n),100)), n=0..8)), k=1..8);
%o (Sage)
%o from mpmath import mp, hyp2f0
%o mp.dps = 25; mp.pretty = True
%o def A247490(k, n):
%o if n < 2: return n
%o if k == 1 and n == 2: return 0 # (failed to converge)
%o return int((-1)^(n+1)*hyp2f0(k, -n+1, 1))
%o for k in (1..8): print([k], [A247490(k, n) for n in (0..8)])
%Y Cf. A000166, A000255, A000153, A000261, A001909, A001910, A176732 - A176736.
%K nonn,tabl
%O 0,13
%A _Peter Luschny_, Sep 20 2014