A332023 - OEIS

%I #16 Mar 07 2021 00:59:12

%S 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,

%T 70,76,83,84,86,89,93,98,104,111,119,120,122,125,129,134,140,147,155,

%U 164,165,167,170,174,179,185,192,200,209,219

%N 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.

%C The sequence increases monotonically.

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

%e The triangle starts:

%e [0]   0;

%e [1]   1,   3;

%e [2]   4,   6,   9;

%e [3]  10,  12,  15,  19;

%e [4]  20,  22,  25,  29,  34;

%e [5]  35,  37,  40,  44,  49,  55;

%e [6]  56,  58,  61,  65,  70,  76,  83;

%e [7]  84,  86,  89,  93,  98, 104, 111, 119;

%e [8] 120, 122, 125, 129, 134, 140, 147, 155, 164;

%e [9] 165, 167, 170, 174, 179, 185, 192, 200, 209, 219;

%p T := (n, k) -> binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1):

%p seq(seq(T(n, k), k=0..n), n=0..9);

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

%Y Cf. A014370.

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Feb 20 2020