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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. 3
 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, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1, 1, 23, 221, 1159, 3649, 7183, 8989, 7183, 3649, 1159, 221, 23, 1, 1, 27, 313, 2047, 8361, 22363, 40081, 48639, 40081 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Rémy Sigrist, Rows n = 0..100 of triangle, flattened Rémy Sigrist, PARI program for A297191 FORMULA From G. C. Greubel, May 25 2021: (Start) T(n, k) = (-1)^k*hypergeometric2F1([-2*n+k, k+1], [1], 2) for 0 <= k <= 2*n. T(n, k) = A008288(2*n, k). Sum_{k=0..2*n} T(n,k) = A000129(2*n+1). (End) EXAMPLE The Delannoy triangle (A008288) begins: 1; 1, 1; 1, 3, 1; 1, 5, 5, 1; 1, 7, 13, 7, 1; 1, 9, 25, 25, 9, 1; 1, 11, 41, 63, 41, 11, 1; 1, 13, 61, 129, 129, 61, 13, 1; 1, 15, 85, 231, 321, 231, 85, 15, 1; 1, 17, 113, 377, 681, 681, 377, 113, 17, 1; this irregular triangle begins: 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, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1; ... MATHEMATICA A297191[n_, k_]:= (-1)^k*Hypergeometric2F1[-2*n+k, k+1, 1, 2]; Table[A297191[n, k], {n, 0, 12}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, May 25 2021 *) PROG (PARI) See Links section. (Sage) def A297191(n, k): return (-1)^k*hypergeometric([-2*n+k, k+1], [1], 2).simplify() flatten([[A297191(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, May 25 2021 CROSSREFS Cf. A000129, A008288, A297192. Sequence in context: A210574 A353532 A049290 * A147990 A134567 A131932 Adjacent sequences: A297188 A297189 A297190 * A297192 A297193 A297194 KEYWORD nonn,tabf,easy AUTHOR N. J. A. Sloane, Jan 10 2018 EXTENSIONS More terms from Rémy Sigrist, Jan 18 2018 STATUS approved

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Last modified May 23 05:11 EDT 2024. Contains 372758 sequences. (Running on oeis4.)