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A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

%I A163770 #29 Aug 22 2025 04:25:41
%S A163770 1,0,1,1,1,2,2,3,4,6,-9,-7,-4,0,6,44,35,28,24,24,30,-165,-121,-86,-58,
%T A163770 -34,-10,20,594,429,308,222,164,130,120,140,-2037,-1443,-1014,-706,
%U A163770 -484,-320,-190,-70,70,6824,4787,3344,2330,1624,1140,820,630,560,630
%N A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).
%C A163770 An analog to the derangement triangle (A068106).
%H A163770 G. C. Greubel, <a href="/A163770/b163770.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163770 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163770 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%H A163770 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.
%F A163770 T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).
%e A163770 1
%e A163770 0, 1
%e A163770 1, 1, 2
%e A163770 2, 3, 4, 6
%e A163770 -9, -7, -4, 0, 6
%e A163770 44, 35, 28, 24, 24, 30
%e A163770 -165, -121, -86, -58, -34, -10, 20
%p A163770 DiffTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
%p A163770 for m from 0 by 1 to n-1 do A[m] := f(m);
%p A163770 for j from m by -1 to 1 do A[j-1] := A[j-1] - A[j] od;
%p A163770 for i from 0 to m do T := T,(-1)^(m-i)*A[i] od;
%p A163770 if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
%p A163770 od; subsop(1=NULL,[T]) end:
%p A163770 swing := proc(n) option remember; if n = 0 then 1 elif
%p A163770 irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A163770 Computes n rows of the triangle.
%p A163770 A163770 := n -> DiffTria(k->swing(k),n,true);
%p A163770 A068106 := n -> DiffTria(k->factorial(k),n,true);
%t A163770 sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2013 *)
%Y A163770 Row sums are A163773.
%Y A163770 Cf. A056040, A163650, A163771, A163772, A068106.
%K A163770 sign,tabl,changed
%O A163770 0,6
%A A163770 _Peter Luschny_, Aug 05 2009