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A178238 Triangle read by rows: partial column sums of the triangle of natural numbers (written sequentially by rows).

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