cp's OEIS Frontend

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.

%I A194586 #14 Jun 26 2019 08:25:43
%S A194586 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 A194586 210,0,630,0,140,0,8,0,336,0,1680,0,1120,0,0,9,0,504,0,3780,0,5040,0,
%U A194586 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 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.
%C A194586 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 A194586 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F A194586 egf(x,y) = x*y*exp(x)*BesselI(0,2*x*y).
%e A194586                0
%e A194586               0, 1
%e A194586             0, 2, 0
%e A194586            0, 3, 0, 6
%e A194586          0, 4, 0, 24, 0
%e A194586        0, 5, 0, 60, 0, 30
%e A194586     0, 6, 0, 120, 0, 180, 0
%e A194586   0, 7, 0, 210, 0, 630, 0, 140
%e A194586                 0
%e A194586                 q
%e A194586                2 q
%e A194586             3 q + 6 q^3
%e A194586            4 q + 24 q^3
%e A194586        5 q + 60 q^3  + 30 q^5
%e A194586       6 q + 120 q^3  + 180 q^5
%e A194586   7 q + 210 q^3  + 630 q^5  + 140 q^7
%p A194586 A194586 := proc(n,k) local j, swing; swing := n -> n!/iquo(n,2)!^2:
%p A194586 add(binomial(n,j)*swing(j)*q^j*(j mod 2),j=0..n); coeff(%,q,k) end:
%p A194586 seq(print(seq(A194586(n,k),k=0..n)),n=0..8);
%t A194586 sf[n_] := n!/Quotient[n, 2]!^2;
%t A194586 row[n_] := Sum[Binomial[n, j] sf[j] q^j Mod[j, 2], {j, 0, n}] // CoefficientList[#, q]& // PadRight[#, n+1]&;
%t A194586 Table[row[n], {n, 0, 12}] (* _Jean-François Alcover_, Jun 26 2019 *)
%Y A194586 Row sums are A109188. Cf. A056040, A005717, A163649, A089627.
%K A194586 nonn,tabl
%O A194586 0,5
%A A194586 _Peter Luschny_, Aug 29 2011