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

1,2

Comments

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

Links

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

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)

Example

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.

Prog

(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

Crossrefs

Column 1 is A004006.

Main diagonal is A000217.

Row sums are A002817.

Cf. A000012, A000027.

Sequence in context: A021300 A121527 A021886 * A255333 A200792 A218567

Adjacent sequences: A178235 A178236 A178237 * A178239 A178240 A178241

Keyword

easy,nonn,tabl

Author

Gary W. Adamson, May 23 2010

Extensions

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

Status

approved