A176123 Irregular triangle, read by rows, T(n, k) = binomial(n-(k-1),k-1), 1 <= k <= floor(n/2-1).

1, 1, 1, 5, 1, 6, 1, 7, 15, 1, 8, 21, 1, 9, 28, 35, 1, 10, 36, 56, 1, 11, 45, 84, 70, 1, 12, 55, 120, 126, 1, 13, 66, 165, 210, 126, 1, 14, 78, 220, 330, 252, 1, 15, 91, 286, 495, 462, 210, 1, 16, 105, 364, 715, 792, 462, 1, 17, 120, 455, 1001, 1287, 924, 330, 1, 18, 136, 560, 1365, 2002, 1716, 792

Offset: 4

Author

Roger L. Bagula, Dec 07 2010

Keywords

• nonn
• tabf

Comments

The row sum is A099572 which has limiting ratio of (1+sqrt(5))/2.

This is Pascal's triangle (A007318) read along upward sloping diagonals and truncated.

Examples

			Triangle begins as:
  1;
  1;
  1,  5;
  1,  6;
  1,  7, 15;
  1,  8, 21;
  1,  9, 28,  35;
  1, 10, 36,  56;
  1, 11, 45,  84,  70;
  1, 12, 55, 120, 126;
  1, 13, 66, 165, 210, 126;
		

Links

• G. C. Greubel, Rows n = 4..100 of triangle, flattened

Crossrefs

Cf. A007318, A099572.

Programs

GAP

Flat(List([4..20], n-> List([1..Int((n-2)/2)], k-> Binomial(n-k+1,k-1) ))); # G. C. Greubel, Nov 27 2019

Magma

[Binomial(n-k+1,k-1): k in [1..Floor((n-2)/2)], n in [4..20]]; // G. C. Greubel, Nov 27 2019

Maple

seq(seq( binomial(n-k+1, k-1), k=1..floor((n-2)/2)), n=4..20); # G. C. Greubel, Nov 27 2019

Mathematica

Table[Binomial[n-k+1, k-1], {n,4,20}, {k, Floor[(n-2)/2]}]//Flatten

PARI

T(n,k) = binomial(n-k+1,k-1);
for(n=4,20, for(k=1, (n-2)\2, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 27 2019

Sage

[[binomial(n-k+1,k-1) for k in (1..floor((n-2)/2))] for n in (4..20)] # G. C. Greubel, Nov 27 2019

Extensions

Edited by N. J. A. Sloane, Dec 09 2010

More terms added by G. C. Greubel, Nov 27 2019