A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).
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, -34, -10, 20, 594, 429, 308, 222, 164, 130, 120, 140, -2037, -1443, -1014, -706, -484, -320, -190, -70, 70, 6824, 4787, 3344, 2330, 1624, 1140, 820, 630, 560, 630
Offset: 0
Examples
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, -34, -10, 20
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.
Programs
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Maple
DiffTria := proc(f,n,display) local m,A,j,i,T; T:=f(0); for m from 0 by 1 to n-1 do A[m] := f(m); for j from m by -1 to 1 do A[j-1] := A[j-1] - A[j] od; for i from 0 to m do T := T,(-1)^(m-i)*A[i] od; if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi; od; subsop(1=NULL,[T]) end: swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end: Computes n rows of the triangle. A163770 := n -> DiffTria(k->swing(k),n,true); A068106 := n -> DiffTria(k->factorial(k),n,true);
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Mathematica
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 *)
Formula
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).
Comments