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A163649
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Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.
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4
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1, -1, 1, 1, -2, 2, -1, 3, -6, 6, 1, -4, 12, -24, 6, -1, 5, -20, 60, -30, 30, 1, -6, 30, -120, 90, -180, 20, -1, 7, -42, 210, -210, 630, -140, 140, 1, -8, 56, -336, 420, -1680, 560, -1120, 70
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OFFSET
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0,5
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COMMENTS
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Given T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!, let A(n,k) = abs(T(n,k)) be the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*q^k. Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the extended Motzkin numbers A189912. (See A089627 for the Motzkin numbers and A194586 for the complementary Motzkin numbers.)
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LINKS
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FORMULA
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T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!.
E.g.f.: egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).
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EXAMPLE
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1
-1, 1
1, -2, 2
-1, 3, -6, 6
1, -4, 12, -24, 6
-1, 5, -20, 60, -30, 30
1, -6, 30, -120, 90, -180, 20
-1, 7, -42, 210, -210, 630, -140, 140
1, -8, 56, -336, 420, -1680, 560, -1120, 70
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MAPLE
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a := proc(n, k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end:
seq(print(seq(a(n, k), k=0..n)), n=0..8);
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MATHEMATICA
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t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ G. C. Greubel, Aug 01 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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