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.

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

Author

Peter Luschny, Aug 29 2011

Keywords

• nonn
• tabl

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.)

Examples

			               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
		

Links

• Peter Luschny, The lost Catalan numbers.

Crossrefs

Row sums are A109188. Cf. A056040, A005717, A163649, A089627.

Programs

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 *)

Formula

egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).