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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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3
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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
| Peter Luschny, The lost Catalan numbers.
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FORMULA
| egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).
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EXAMPLE
| 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
| 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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CROSSREFS
| Row sums give A163650, row sums of absolute values give A163865.
Aerated versions A194586 (odd case) and A089627 (even case).
Sequence in context: A158825 A107111 A082037 * A110858 A008279 A056043
Adjacent sequences: A163646 A163647 A163648 * A163650 A163651 A163652
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KEYWORD
| sign,tabl
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Aug 02 2009
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