A330798 Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.

1, 2, 2, 6, 15, 9, 20, 84, 112, 48, 70, 420, 900, 825, 275, 252, 1980, 5940, 8580, 6006, 1638, 924, 9009, 35035, 70070, 76440, 43316, 9996, 3432, 40040, 192192, 495040, 742560, 651168, 310080, 62016, 12870, 175032, 1002456, 3174444, 6104700, 7325640, 5372136, 2206413, 389367

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

Cf. A000984, A033184, A174687, A330801.

Sequence in context: A071208 A231536 A216242 * A260687 A083555 A233147

Adjacent sequences: A330795 A330796 A330797 * A330799 A330800 A330801

Keyword

nonn,tabl

Author

Peter Luschny, Jan 02 2020

Status

approved