OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) := ((n+1)/(2*n+1))*binomial(2*n+1, n+k+1)*binomial(2*n+k, k).
T(n, 0) = A000984(n).
T(n, n) = A174687(n).
Sum_{k=0..n} T(n, k) = A330801(n).
Sum_{k=0..n} (-1)^k*T(n, k) = 0^n. - G. C. Greubel, May 23 2023
EXAMPLE
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7]
[0] 1
[1] 2, 2
[2] 6, 15, 9
[3] 20, 84, 112, 48
[4] 70, 420, 900, 825, 275
[5] 252, 1980, 5940, 8580, 6006, 1638
[6] 924, 9009, 35035, 70070, 76440, 43316, 9996
[7] 3432, 40040, 192192, 495040, 742560, 651168, 310080, 6201
MAPLE
alias(C=binomial): T := (n, k) -> ((n+1)/(2*n+1))*C(2*n+1, n+k+1)*C(2*n+k, k):
seq(seq(T(n, k), k=0..n), n=0..8);
MATHEMATICA
T[n_, k_]:= ((n+1)/(n+k+1))*Binomial[n, k]*Binomial[2*n+k, n];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 23 2023 *)
PROG
(Magma)
A330798:= func< n, k | ((n+1)/(n+k+1))*Binomial(n, k)*Binomial(2*n+k, n) >;
[A330798(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 23 2023
(SageMath)
def A330798(n, k): return ((n+1)/(n+k+1))*binomial(n, k)*binomial(2*n+k, n)
flatten([[A330798(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 23 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 02 2020
STATUS
approved