A166454 Triangle read by rows: T(n, k) = (1/2)*(A007318(n,k) - A047999(n,k)).

1, 1, 1, 2, 3, 2, 2, 5, 5, 2, 3, 7, 10, 7, 3, 3, 10, 17, 17, 10, 3, 4, 14, 28, 35, 28, 14, 4, 4, 18, 42, 63, 63, 42, 18, 4, 5, 22, 60, 105, 126, 105, 60, 22, 5, 5, 27, 82, 165, 231, 231, 165, 82, 27, 5, 6, 33, 110, 247, 396, 462, 396, 247, 110, 33, 6

Offset: 2

Author

Gary W. Adamson, Oct 14 2009

Keywords

• nonn
• tabl

Comments

Row sums = A120739: (1, 2, 7, 14, 30, 60, 127, 254, ...).

Examples

			First few rows of the triangle:
  1;
  1,   1;
  2,   3,   2;
  2,   5,   5,   2;
  3,   7,  10,   7,   3;
  3,  10,  17,  17,  10,   3;
  4,  14,  28,  35,  28,  14,   4;
  4,  18,  42,  63,  63,  42,  18,   4;
  5,  22,  60, 105, 126, 105,  60,  22,   5;
  5,  27,  82, 165, 231, 231, 165,  82,  27,   5;
  6,  33, 110, 247, 396, 462, 396, 247, 110,  33,   6;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 2..125 of triangle, flattened

Crossrefs

Cf. A047999, A120739.

Cf. A007318, A011848, A001700 (central terms).

Programs

GAP

Flat(List([2..12],n->List([1..n-1],m->Int(Binomial(n,m)/2)))); # Muniru A Asiru, Apr 14 2019

Haskell

Following Bagula's formula
a166454 n k = a166454_tabl !! (n-2) !! (k-1)
a166454_row n = a166454_tabl !! (n-2)
a166454_tabl = map (map (flip div 2) . init . tail) $ drop 2 a007318_tabl
-- Reinhard Zumkeller, Mar 04 2015

Magma

[[Floor(Binomial(n,k)/2): k in [1..n-1]]: n in [2..12]]; // G. C. Greubel, Apr 16 2019

Maple

seq(seq(floor(binomial(n,m)/2),m=1..n-1),n=2..12); # Muniru A Asiru, Apr 14 2019

Mathematica

T[n_, m_] = Floor[Binomial[n, m]/2]; Table[T[n, m], {n, 2, 12}, {m, 1, n-1}]//Flatten (* Roger L. Bagula, Mar 07 2010*)

PARI

{T(n,k) = binomial(n,k)\2 };
for(n=2,12, for(k=1,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 16 2019

Sage

[[floor(binomial(n,k)/2) for k in (1..n-1)] for n in (2..12)] # G. C. Greubel, Apr 16 2019

Formula

T(n, k) = (1/2)*(A007318(n,k) - A047999(n,k)), nonzero terms.

T(n, m) = floor(binomial(n, m)/2). - Roger L. Bagula, Mar 07 2010