%I #14 Jun 26 2019 08:25:43

%S 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,

%T 210,0,630,0,140,0,8,0,336,0,1680,0,1120,0,0,9,0,504,0,3780,0,5040,0,

%U 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

%N 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.

%C 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.)

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.

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

%e 0

%e 0, 1

%e 0, 2, 0

%e 0, 3, 0, 6

%e 0, 4, 0, 24, 0

%e 0, 5, 0, 60, 0, 30

%e 0, 6, 0, 120, 0, 180, 0

%e 0, 7, 0, 210, 0, 630, 0, 140

%e 0

%e q

%e 2 q

%e 3 q + 6 q^3

%e 4 q + 24 q^3

%e 5 q + 60 q^3 + 30 q^5

%e 6 q + 120 q^3 + 180 q^5

%e 7 q + 210 q^3 + 630 q^5 + 140 q^7

%p A194586 := proc(n,k) local j, swing; swing := n -> n!/iquo(n,2)!^2:

%p add(binomial(n,j)*swing(j)*q^j*(j mod 2),j=0..n); coeff(%,q,k) end:

%p seq(print(seq(A194586(n,k),k=0..n)),n=0..8);

%t sf[n_] := n!/Quotient[n, 2]!^2;

%t row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;

%t Table[row[n], {n, 0, 12}] (* _Jean-François Alcover_, Jun 26 2019 *)

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

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Aug 29 2011