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A247490 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).

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