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

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, 61, 48, 28, 92, 98, 101, 100, 94, 82, 63, 36, 129, 136, 140, 140, 135, 124, 106, 80, 45, 175, 183, 188, 189, 185, 175, 158, 133, 99, 55, 231, 240, 246, 248, 245, 236, 220, 196, 163, 120, 66

Offset: 1

Author

Gary W. Adamson, May 23 2010

Keywords

• easy
• nonn
• tabl

Comments

T(n,k) is the n-th partial sum of the k-th column of the triangle of natural numbers.

Examples

			First few rows of the triangle:
    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,  61,  48,  28;
   92,  98, 101, 100,  94,  82,  63,  36;
  129, 136, 140, 140, 135, 124, 106,  80,  45;
  175, 183, 188, 189, 185, 175, 158, 133,  99,  55;
  231, 240, 246, 248, 245, 236, 220, 196, 163, 120,  66;
  298, 308, 314, 318, 316, 308, 293, 270, 238, 196, 143, 78;
  ...
These are the partial sums of the columns of the triangle:
  1;
  2, 3;
  4, 5, 6;
  7, 8, 9, 10;
  ...
For example, T(4,2) = 3 + 5 + 8 = 16.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Column 1 is A004006.

Main diagonal is A000217.

Row sums are A002817.

Cf. A000012, A000027.

Programs

PARI

T(n,k) = {binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1)}
{ for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) } \\ Andrew Howroyd, Apr 18 2021

Formula

As infinite lower triangular matrices, A000012 * A000027.

From Andrew Howroyd, Apr 18 2021: (Start)

T(n,k) = Sum_{j=k..n} (k + j*(j-1)/2).

T(n,k) = binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1).

T(2*n, n) = A255211(n).

(End)

Extensions

Name changed and terms a(56) and beyond from Andrew Howroyd, Apr 18 2021