A178238 - OEIS

%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