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

4,4

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.

Links

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

Example

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;

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

Prog

(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
(Magma) [Binomial(n-k+1, k-1): k in [1..Floor((n-2)/2)], n in [4..20]]; // 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
(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

Crossrefs

Cf. A007318, A099572.

Sequence in context: A176320 A190185 A331189 * A066805 A326142 A028284

Adjacent sequences: A176120 A176121 A176122 * A176124 A176125 A176126

Keyword

nonn,tabf

Author

Roger L. Bagula, Dec 07 2010

Extensions

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

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

Status

approved