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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 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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