OFFSET
0,3
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)$ where i$ denotes the swinging factorial of i (A056040).
This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019
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
Triangle begins
1;
1, 2;
3, 4, 6;
7, 10, 14, 20;
19, 26, 36, 50, 70;
51, 70, 96, 132, 182, 252;
141, 192, 262, 358, 490, 672, 924;
From M. F. Hasler, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
1 1 3 7 19 51 ...
2 4 10 26 70 192 ...
6 14 36 96 262 726 ...
20 50 132 358 988 2760 ...
70 182 490 1346 3748 10540 ...
252 672 1836 5094 14288 40404 ...
(...)
Read by falling antidiagonals this yields the same sequence. (End)
MAPLE
For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k), 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], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 05 2009
STATUS
approved