A341103 - OEIS

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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

(list; table; graph; refs; listen; history; text; internal format)

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