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%I #8 Apr 30 2021 01:03:01
%S 1,3,3,7,8,6,14,16,15,10,25,28,28,24,15,41,45,46,43,35,21,63,68,70,68,
%T 61,48,28,92,98,101,100,94,82,63,36,129,136,140,140,135,124,106,80,45,
%U 175,183,188,189,185,175,158,133,99,55,231,240,246,248,245,236,220,196,163,120,66
%N Triangle read by rows: partial column sums of the triangle of natural numbers (written sequentially by rows).
%C T(n,k) is the n-th partial sum of the k-th column of the triangle of natural numbers.
%H Andrew Howroyd, <a href="/A178238/b178238.txt">Table of n, a(n) for n = 1..1275</a>
%F As infinite lower triangular matrices, A000012 * A000027.
%F From _Andrew Howroyd_, Apr 18 2021: (Start)
%F T(n,k) = Sum_{j=k..n} (k + j*(j-1)/2).
%F T(n,k) = binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1).
%F T(2*n, n) = A255211(n).
%F (End)
%e First few rows of the triangle:
%e 1;
%e 3, 3;
%e 7, 8, 6;
%e 14, 16, 15, 10;
%e 25, 28, 28, 24, 15;
%e 41, 45, 46, 43, 35, 21;
%e 63, 68, 70, 68, 61, 48, 28;
%e 92, 98, 101, 100, 94, 82, 63, 36;
%e 129, 136, 140, 140, 135, 124, 106, 80, 45;
%e 175, 183, 188, 189, 185, 175, 158, 133, 99, 55;
%e 231, 240, 246, 248, 245, 236, 220, 196, 163, 120, 66;
%e 298, 308, 314, 318, 316, 308, 293, 270, 238, 196, 143, 78;
%e ...
%e These are the partial sums of the columns of the triangle:
%e 1;
%e 2, 3;
%e 4, 5, 6;
%e 7, 8, 9, 10;
%e ...
%e For example, T(4,2) = 3 + 5 + 8 = 16.
%o (PARI) T(n,k) = {binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1)}
%o { for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) } \\ _Andrew Howroyd_, Apr 18 2021
%Y Column 1 is A004006.
%Y Main diagonal is A000217.
%Y Row sums are A002817.
%Y Cf. A000012, A000027.
%K easy,nonn,tabl
%O 1,2
%A _Gary W. Adamson_, May 23 2010
%E Name changed and terms a(56) and beyond from _Andrew Howroyd_, Apr 18 2021
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