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A163770
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Triangle interpolating the swinging sub-factorial (A163650) with the swinging factorial (A056040). An analog to the derangement triangle (A068106).
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4
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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; internal format)
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OFFSET
| 0,6
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COMMENTS
| 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)*i$
where i$ denotes the swinging factorial of i (A056040).
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REFERENCES
| Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.
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LINKS
| M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Peter Luschny, Swinging Factorial.
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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
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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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CROSSREFS
| Sum rows are A163773. Cf. A056040, 163650, A163771, A163772, A068106.
Sequence in context: A017912 A102543 A068598 * A035561 A068106 A186964
Adjacent sequences: A163767 A163768 A163769 * A163771 A163772 A163773
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KEYWORD
| sign,tabl
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Aug 05 2009
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