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URL: https://oeis.org/A184883

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A184883
Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).
4
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
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
(SageMath)
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
KEYWORD
nonn,easy,tabl
AUTHOR
Paul Barry, Jan 24 2011
STATUS
approved