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A163770 Triangle interpolating the swinging sub-factorial (A163650) with the swinging factorial (A056040). An analog to the derangement triangle (A068106). 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

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).

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

Table of n, a(n) for n=0..54.

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.

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-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Row sums are A163773. Cf. A056040, A163650, A163771, A163772, A068106.

Sequence in context: A017912 A102543 A068598 * 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 April 25 19:05 EDT 2017. Contains 285426 sequences.