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A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

%I A163840 #25 Aug 22 2025 04:25:26
%S A163840 1,2,1,5,3,2,16,11,8,6,47,31,20,12,6,146,99,68,48,36,30,447,301,202,
%T A163840 134,86,50,20,1380,933,632,430,296,210,160,140,4251,2871,1938,1306,
%U A163840 876,580,370,210,70,13102,8851,5980,4042,2736,1860,1280,910,700,630
%N A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).
%C A163840 Triangle read by rows.
%C A163840 An analog to the binomial triangle of the factorials (A076571).
%H A163840 G. C. Greubel, <a href="/A163840/b163840.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A163840 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163840 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163840 T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040), for n >= 0, k >= 0.
%e A163840 Triangle begins
%e A163840     1;
%e A163840     2,   1;
%e A163840     5,   3,   2;
%e A163840    16,  11,   8,   6;
%e A163840    47,  31,  20,  12,  6;
%e A163840   146,  99,  68,  48, 36, 30;
%e A163840   447, 301, 202, 134, 86, 50, 20;
%p A163840 SumTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
%p A163840 for m from 0 by 1 to n-1 do A[m] := f(m);
%p A163840 for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
%p A163840 for i from 0 to m do T := T,A[i] od;
%p A163840 if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
%p A163840 od; subsop(1=NULL,[T]) end:
%p A163840 swing := proc(n) option remember; if n = 0 then 1 elif
%p A163840 irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A163840 # Computes n rows of the triangle:
%p A163840 A163840 := n -> SumTria(swing,n,true);
%t A163840 sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[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 A163840 Row sums are A163843.
%Y A163840 Cf. A056040, A163865, A163841, A163842, A163650.
%K A163840 nonn,tabl,changed
%O A163840 0,2
%A A163840 _Peter Luschny_, Aug 06 2009