login
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
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
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Feb 24 2021
STATUS
approved