A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).

1, 1, 1, 1, 5, 1, 1, 9, 13, 1, 1, 13, 41, 25, 1, 1, 17, 85, 129, 41, 1, 1, 21, 145, 377, 321, 61, 1, 1, 25, 221, 833, 1289, 681, 85, 1, 1, 29, 313, 1561, 3649, 3653, 1289, 113, 1, 1, 33, 421, 2625, 8361, 13073, 8989, 2241, 145, 1, 1, 37, 545, 4089, 16641, 36365, 40081, 19825, 3649, 181, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n,k) = [k<=n]*Sum_{j=0..k} C(2*n-2*k,j)*C(k,j)*2^j.

T(n, n-k) = A114123(n, k).

Sum_{k=0..n} T(n, k) = A099463(n+1).

Sum_{k=0..floor(n/2)} T(n, k) = A184884(n).

T(n, k) = Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - G. C. Greubel, Nov 19 2021

Example

Triangle begins
  1;
  1,  1;
  1,  5,   1;
  1,  9,  13,    1;
  1, 13,  41,   25,     1;
  1, 17,  85,  129,    41,     1;
  1, 21, 145,  377,   321,    61,     1;
  1, 25, 221,  833,  1289,   681,    85,     1;
  1, 29, 313, 1561,  3649,  3653,  1289,   113,    1;
  1, 33, 421, 2625,  8361, 13073,  8989,  2241,  145,   1;
  1, 37, 545, 4089, 16641, 36365, 40081, 19825, 3649, 181, 1;

Mathematica

A184883[n_, k_]:= Hypergeometric2F1[-k, 2*(k-n), 1, 2];
Table[A184883[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 19 2021 *)

Prog

(Magma)
T:= func< n, k | (&+[Binomial(k, j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2021
(Sage)
def A184883(n, k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
flatten([[A184883(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 19 2021

Crossrefs

Cf. A099463 (row sums), A114123, A184884 (diagonal sums).

Sequence in context: A131061 A157169 A081578 * A279003 A210651 A255831

Adjacent sequences: A184880 A184881 A184882 * A184884 A184885 A184886

Keyword

nonn,easy,tabl

Author

Paul Barry, Jan 24 2011

Status

approved