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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
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
Sequence in context: A210574 A353532 A049290 * A147990 A134567 A131932
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.)