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A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457). 0
1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Triangle read by rows.

For n >= 0, k >= 0 let T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ 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..44.

Peter Luschny, Swinging Factorial.

FORMULA

Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - Peter Bala, Nov 10 2013

EXAMPLE

1

-1, 6

1, -12, 30

-1, 18, -90, 140

1, -24, 180, -560, 630

-1, 30, -300, 1400, -3150, 2772

1, -36, 450, -2800, 9450, -16632, 12012

MAPLE

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:

a := proc(n, k) (-1)^(n-k)*binomial(n, k)*swing(2*k+1) end:

seq(print(seq(a(n, k), k=0..n)), n=0..8);

CROSSREFS

Row sums are the inverse binomial transform of the beta numbers (A163872).

Cf. A163649, A098473, A056040.

Sequence in context: A162933 A229085 A090850 * A013613 A122508 A171006

Adjacent sequences:  A163942 A163943 A163944 * A163946 A163947 A163948

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Aug 07 2009

STATUS

approved

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Last modified June 24 11:32 EDT 2017. Contains 288697 sequences.