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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.

%I A332023 #16 Mar 07 2021 00:59:12
%S A332023 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 A332023 70,76,83,84,86,89,93,98,104,111,119,120,122,125,129,134,140,147,155,
%U A332023 164,165,167,170,174,179,185,192,200,209,219
%N 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.
%C A332023 The sequence increases monotonically.
%F A332023 T(n, k) = (1/6)*(3*k^2 + 9*k + n*(n + 1)*(n + 2)).
%e A332023 The triangle starts:
%e A332023 [0]   0;
%e A332023 [1]   1,   3;
%e A332023 [2]   4,   6,   9;
%e A332023 [3]  10,  12,  15,  19;
%e A332023 [4]  20,  22,  25,  29,  34;
%e A332023 [5]  35,  37,  40,  44,  49,  55;
%e A332023 [6]  56,  58,  61,  65,  70,  76,  83;
%e A332023 [7]  84,  86,  89,  93,  98, 104, 111, 119;
%e A332023 [8] 120, 122, 125, 129, 134, 140, 147, 155, 164;
%e A332023 [9] 165, 167, 170, 174, 179, 185, 192, 200, 209, 219;
%p A332023 T := (n, k) -> binomial(n+2, 3) + binomial(k+1, 2) + binomial(k, 1):
%p A332023 seq(seq(T(n, k), k=0..n), n=0..9);
%Y A332023 Cf. A000292 (first column), A062748 (diagonal), A005286 (subdiagonal), A332697 (row sums).
%Y A332023 Cf. A014370.
%K A332023 nonn,tabl
%O A332023 0,3
%A A332023 _Peter Luschny_, Feb 20 2020