%I #32 Aug 05 2024 12:05:54

%S 1,1,2,3,4,6,7,10,14,20,19,26,36,50,70,51,70,96,132,182,252,141,192,

%T 262,358,490,672,924,393,534,726,988,1346,1836,2508,3432,1107,1500,

%U 2034,2760,3748,5094,6930,9438,12870

%N Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).

%C 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).

%C 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

%H G. C. Greubel, <a href="/A163771/b163771.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.

%H Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.

%H M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

%e Triangle begins

%e 1;

%e 1, 2;

%e 3, 4, 6;

%e 7, 10, 14, 20;

%e 19, 26, 36, 50, 70;

%e 51, 70, 96, 132, 182, 252;

%e 141, 192, 262, 358, 490, 672, 924;

%e From _M. F. Hasler_, Nov 15 2019: (Start)

%e The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:

%e 1 1 3 7 19 51 ...

%e 2 4 10 26 70 192 ...

%e 6 14 36 96 262 726 ...

%e 20 50 132 358 988 2760 ...

%e 70 182 490 1346 3748 10540 ...

%e 252 672 1836 5094 14288 40404 ...

%e (...)

%e Read by falling antidiagonals this yields the same sequence. (End)

%p For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.

%p a := n -> DiffTria(k->swing(2*k),n,true);

%t 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 *)

%Y Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Aug 05 2009