A163842 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869). Triangle read by rows, for n >= 0, k >= 0.

1, 7, 6, 43, 36, 30, 249, 206, 170, 140, 1395, 1146, 940, 770, 630, 7653, 6258, 5112, 4172, 3402, 2772, 41381, 33728, 27470, 22358, 18186, 14784, 12012, 221399, 180018, 146290, 118820, 96462, 78276, 63492, 51480

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• tabl
• changed

Examples

			Triangle begins:
      1;
      7,     6;
     43,    36,    30;
    249,   206,   170,   140;
   1395,  1146,   940,   770,   630;
   7653,  6258,  5112,  4172,  3402,  2772;
  41381, 33728, 27470, 22358, 18186, 14784, 12012;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Factorial.

Crossrefs

Row sums are A163845. Cf. A056040, A163650, A163841, A163842, A163840, A002426, A000984.

Programs

Maple

# Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k+1),n,true);

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

Formula

T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).