A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse.

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

0,2

Comments

Triangle read by rows. For n >= 0, k >= 0 let

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).

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.

Example

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

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 *)

Crossrefs

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

Sequence in context: A240400 A031448 A290254 * A056509 A129722 A133608

Adjacent sequences: A163769 A163770 A163771 * A163773 A163774 A163775

Keyword

nonn,tabl

Author

Peter Luschny, Aug 05 2009

Status

approved