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).

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

Offset

0,2

Comments

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

T(n,k) = sum{i=k..n} 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.

Example

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

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 *)

Crossrefs

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

Sequence in context: A163260 A073112 A070425 * A328697 A288245 A038272

Adjacent sequences: A163839 A163840 A163841 * A163843 A163844 A163845

Keyword

nonn,tabl

Author

Peter Luschny, Aug 06 2009

Status

approved