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A194586
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.
2
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
OFFSET
0,5
COMMENTS
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.)
FORMULA
egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).
EXAMPLE
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
MAPLE
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);
MATHEMATICA
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]&;
Table[row[n], {n, 0, 12}] (* Jean-François Alcover, Jun 26 2019 *)
CROSSREFS
Row sums are A109188. Cf. A056040, A005717, A163649, A089627.
Sequence in context: A298203 A298209 A211871 * A288437 A287736 A180969
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 29 2011
STATUS
approved