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A194586
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Triangle read by rows, T(n,k) the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*(k mod 2)*q^k.
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2
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0, 0, 1, 0, 2, 0, 0, 3, 0, 6, 0, 4, 0, 24, 0, 0, 5, 0, 60, 0, 30, 0, 6, 0, 120, 0, 180, 0, 0, 7, 0, 210, 0, 630, 0, 140, 0, 8, 0, 336, 0, 1680, 0, 1120, 0, 0, 9, 0, 504, 0, 3780, 0, 5040, 0, 630, 0, 10, 0, 720, 0, 7560, 0, 16800, 0, 6300, 0, 0, 11, 0, 990, 0, 13860, 0, 46200, 0, 34650, 0, 2772, 0, 12
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OFFSET
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0,5
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COMMENTS
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Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the complementary Motzkin numbers A005717. (See A089627 for the Motzkin numbers and A163649 for the extended Motzkin numbers.)
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LINKS
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FORMULA
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egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).
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EXAMPLE
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0
0, 1
0, 2, 0
0, 3, 0, 6
0, 4, 0, 24, 0
0, 5, 0, 60, 0, 30
0, 6, 0, 120, 0, 180, 0
0, 7, 0, 210, 0, 630, 0, 140
0
q
2 q
3 q + 6 q^3
4 q + 24 q^3
5 q + 60 q^3 + 30 q^5
6 q + 120 q^3 + 180 q^5
7 q + 210 q^3 + 630 q^5 + 140 q^7
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MAPLE
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A194586 := proc(n, k) local j, swing; swing := n -> n!/iquo(n, 2)!^2:
add(binomial(n, j)*swing(j)*q^j*(j mod 2), j=0..n); coeff(%, q, k) end:
seq(print(seq(A194586(n, k), k=0..n)), n=0..8);
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MATHEMATICA
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sf[n_] := n!/Quotient[n, 2]!^2;
row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;
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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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