cp's OEIS Frontend

A163771 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).

%I A163771 #32 Aug 05 2024 12:05:54
%S A163771 1,1,2,3,4,6,7,10,14,20,19,26,36,50,70,51,70,96,132,182,252,141,192,
%T A163771 262,358,490,672,924,393,534,726,988,1346,1836,2508,3432,1107,1500,
%U A163771 2034,2760,3748,5094,6930,9438,12870
%N A163771 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 A163771 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 A163771 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 A163771 G. C. Greubel, <a href="/A163771/b163771.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163771 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163771 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%H A163771 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 A163771 Triangle begins
%e A163771     1;
%e A163771     1,   2;
%e A163771     3,   4,   6;
%e A163771     7,  10,  14,  20;
%e A163771    19,  26,  36,  50,  70;
%e A163771    51,  70,  96, 132, 182, 252;
%e A163771   141, 192, 262, 358, 490, 672, 924;
%e A163771 From _M. F. Hasler_, Nov 15 2019: (Start)
%e A163771 The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
%e A163771      1   1    3    7    19    51 ...
%e A163771      2   4   10   26    70   192 ...
%e A163771      6  14   36   96   262   726 ...
%e A163771     20  50  132  358   988  2760 ...
%e A163771     70 182  490 1346  3748 10540 ...
%e A163771    252 672 1836 5094 14288 40404 ...
%e A163771   (...)
%e A163771 Read by falling antidiagonals this yields the same sequence. (End)
%p A163771 For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
%p A163771 a := n -> DiffTria(k->swing(2*k),n,true);
%t A163771 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 A163771 Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.
%K A163771 nonn,tabl
%O A163771 0,3
%A A163771 _Peter Luschny_, Aug 05 2009