A341103 T(n, k) = Sum_{j=0..k}(binomial(n + k - j, 2*k) - binomial(n + j - 1, 2*k)) for n > 0 and T(0, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 1, 0, 3, 1, 0, 5, 5, 1, 0, 7, 15, 7, 1, 0, 9, 34, 28, 9, 1, 0, 11, 65, 84, 45, 11, 1, 0, 13, 111, 209, 165, 66, 13, 1, 0, 15, 175, 455, 495, 286, 91, 15, 1, 0, 17, 260, 896, 1286, 1001, 455, 120, 17, 1, 0, 19, 369, 1632, 2994, 3003, 1820, 680, 153, 19, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

T(2*n, n) = binomial(3*n, n).

Example

Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 3,  1]
[3] [0, 5,  5,   1]
[4] [0, 7,  15,  7,   1]
[5] [0, 9,  34,  28,  9,    1]
[6] [0, 11, 65,  84,  45,   11,   1]
[7] [0, 13, 111, 209, 165,  66,   13,  1]
[8] [0, 15, 175, 455, 495,  286,  91,  15,  1]
[9] [0, 17, 260, 896, 1286, 1001, 455, 120, 17, 1]

Prog

(SageMath)
def T(n, k):
    if k == 0: return 0^n
    return sum(binomial(n + k - j, 2*k) - binomial(n + j - 1, 2*k)
           for j in (0..k))
for n in range(11):
    print([T(n, k) for k in (0..n)])

Crossrefs

Cf. A341104 (row sums), A005809 (central terms).

Sequence in context: A324664 A011084 A347924 * A021326 A362885 A227342

Adjacent sequences: A341100 A341101 A341102 * A341104 A341105 A341106

Keyword

nonn,tabl

Author

Peter Luschny, Feb 24 2021

Status

approved