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A297191 Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.

%I A297191 #16 May 26 2021 02:16:54
%S A297191 1,1,3,1,1,7,13,7,1,1,11,41,63,41,11,1,1,15,85,231,321,231,85,15,1,1,
%T A297191 19,145,575,1289,1683,1289,575,145,19,1,1,23,221,1159,3649,7183,8989,
%U A297191 7183,3649,1159,221,23,1,1,27,313,2047,8361,22363,40081,48639,40081
%N A297191 Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.
%H A297191 Rémy Sigrist, <a href="/A297191/b297191.txt">Rows n = 0..100 of triangle, flattened</a>
%H A297191 Rémy Sigrist, <a href="/A297191/a297191.gp.txt">PARI program for A297191</a>
%F A297191 From _G. C. Greubel_, May 25 2021: (Start)
%F A297191 T(n, k) = (-1)^k*hypergeometric2F1([-2*n+k, k+1], [1], 2) for 0 <= k <= 2*n.
%F A297191 T(n, k) = A008288(2*n, k).
%F A297191 Sum_{k=0..2*n} T(n,k) = A000129(2*n+1). (End)
%e A297191 The Delannoy triangle (A008288) begins:
%e A297191   1;
%e A297191   1,  1;
%e A297191   1,  3,   1;
%e A297191   1,  5,   5,   1;
%e A297191   1,  7,  13,   7,   1;
%e A297191   1,  9,  25,  25,   9,   1;
%e A297191   1, 11,  41,  63,  41,  11,   1;
%e A297191   1, 13,  61, 129, 129,  61,  13,   1;
%e A297191   1, 15,  85, 231, 321, 231,  85,  15,  1;
%e A297191   1, 17, 113, 377, 681, 681, 377, 113, 17, 1;
%e A297191 this irregular triangle begins:
%e A297191   1;
%e A297191   1,  3,   1;
%e A297191   1,  7,  13,   7,    1;
%e A297191   1, 11,  41,  63,   41,   11,    1;
%e A297191   1, 15,  85, 231,  321,  231,   85,  15,   1;
%e A297191   1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1;
%e A297191   ...
%t A297191 A297191[n_, k_]:= (-1)^k*Hypergeometric2F1[-2*n+k, k+1, 1, 2];
%t A297191 Table[A297191[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, May 25 2021 *)
%o A297191 (PARI) See Links section.
%o A297191 (Sage)
%o A297191 def A297191(n,k): return (-1)^k*hypergeometric([-2*n+k, k+1], [1], 2).simplify()
%o A297191 flatten([[A297191(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, May 25 2021
%Y A297191 Cf. A000129, A008288, A297192.
%K A297191 nonn,tabf,easy
%O A297191 0,3
%A A297191 _N. J. A. Sloane_, Jan 10 2018
%E A297191 More terms from _Rémy Sigrist_, Jan 18 2018