A332023 T(n, k) = binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1). Triangle read by rows, T(n, k) for 0 <= k <= n.

0, 1, 3, 4, 6, 9, 10, 12, 15, 19, 20, 22, 25, 29, 34, 35, 37, 40, 44, 49, 55, 56, 58, 61, 65, 70, 76, 83, 84, 86, 89, 93, 98, 104, 111, 119, 120, 122, 125, 129, 134, 140, 147, 155, 164, 165, 167, 170, 174, 179, 185, 192, 200, 209, 219

Offset: 0

Author

Peter Luschny, Feb 20 2020

Keywords

• nonn
• tabl

Comments

The sequence increases monotonically.

Examples

			The triangle starts:
[0]   0;
[1]   1,   3;
[2]   4,   6,   9;
[3]  10,  12,  15,  19;
[4]  20,  22,  25,  29,  34;
[5]  35,  37,  40,  44,  49,  55;
[6]  56,  58,  61,  65,  70,  76,  83;
[7]  84,  86,  89,  93,  98, 104, 111, 119;
[8] 120, 122, 125, 129, 134, 140, 147, 155, 164;
[9] 165, 167, 170, 174, 179, 185, 192, 200, 209, 219;
		

Crossrefs

Cf. A000292 (first column), A062748 (diagonal), A005286 (subdiagonal), A332697 (row sums).

Cf. A014370.

Programs

Maple

T := (n, k) -> binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1):
seq(seq(T(n, k), k=0..n), n=0..9);

Formula

T(n, k) = (1/6)*(3*k^2 + 9*k + n*(n + 1)*(n + 2)).