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 A163770 Triangle read by rows interpolating the swinging sub-factorial (A163650) with the swinging factorial (A056040). 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS An analog to the derangement triangle (A068106). REFERENCES Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened 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. 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). EXAMPLE 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 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); 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 *) CROSSREFS Row sums are A163773. Cf. A056040, A163650, A163771, A163772, A068106. Sequence in context: A316076 A068598 A293165 * A035561 A068106 A186964 Adjacent sequences:  A163767 A163768 A163769 * A163771 A163772 A163773 KEYWORD sign,tabl AUTHOR Peter Luschny, Aug 05 2009 STATUS approved

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Last modified November 17 22:28 EST 2018. Contains 317279 sequences. (Running on oeis4.)