OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
T(n, k) = Sum_{j = k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1). - Detlef Meya, Jan 15 2024
EXAMPLE
Triangle starts:
[0] 1
[1] 5, 1
[2] 45, 10, 1
[3] 505, 115, 15, 1
[4] 6345, 1460, 210, 20, 1
[5] 85405, 19765, 2990, 330, 25, 1
[6] 1204245, 279710, 43635, 5220, 475, 30, 1
MATHEMATICA
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
T[n_, k_] := Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}];
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Jan 15 2024 *)
PROG
(PARI) T(n, k) = sum(j = k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1)) \\ Andrew Howroyd, Jan 15 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 08 2018
STATUS
approved