A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. Triangle read by rows. For n >= 0, k >= 0.

1, 5, 6, 19, 24, 30, 67, 86, 110, 140, 227, 294, 380, 490, 630, 751, 978, 1272, 1652, 2142, 2772, 2445, 3196, 4174, 5446, 7098, 9240, 12012, 7869, 10314, 13510, 17684, 23130, 30228, 39468, 51480

Offset: 0

Author

Peter Luschny, Aug 05 2009

Keywords

• nonn
• tabl
• changed

Examples

			Triangle begins:
     1;
     5,    6;
    19,   24,   30;
    67,   86,  110,  140;
   227,  294,  380,  490,  630;
   751,  978, 1272, 1652, 2142, 2772;
  2445, 3196, 4174, 5446, 7098, 9240, 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.
• M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Crossrefs

Row sums are A163775. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

Programs

Maple

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k+1),n,true);

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[ (-1)^(n-i)*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} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).