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

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

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