A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).

1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• tabl
• changed

Comments

For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040). Triangle read by rows.

Examples

			Triangle begins
     1;
     3,    2;
    11,    8,    6;
    45,   34,   26,   20;
   195,  150,  116,   90,   70;
   873,  678,  528,  412,  322,  252;
  3989, 3116, 2438, 1910, 1498, 1176,  924;
		

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.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, 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),n,true);

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)