On the coefficients of Psi(x,0) of the unique solution of the functional equ\

                             2
    ation, (Psi(x, 0) x y - y  + x - y) Psi(x, y) + y - x Psi(x, 0) = 0



                              By Shalosh B. Ekhad



      Let's g(x)=Psi(x,0). The first, 50, (for the sake of the OEIS) are



[1, 1, 2, 3, 7, 12, 30, 55, 143, 273, 728, 1428, 3876, 7752, 21318, 43263, 
120175, 246675, 690690, 1430715, 4032015, 8414640, 23841480, 50067108, 
142498692, 300830572, 859515920, 1822766520, 5225264024, 11124755664, 
31983672534, 68328754959, 196947587823, 422030545335, 1219199353190, 
2619631042665, 7583142491925, 16332922290300, 47365474641870, 102240109897695,
296983176369495, 642312451217745, 1868545312633440, 4048514844039120, 
11793499763070480, 25594403741131680, 74650344244967400, 162250238001816900, 
473770694965305348, 1031147983159782228]


               g=g(x) satisfies the following algebraic equation



                            3        2
                           g  x - 2 g  + 3 g - 1 = 0



                             and in Maple notation



g^3*x-2*g^2+3*g-1 = 0


        More generally, psi=Psi(x,y) satisisfies the algebraic equation



   3  6      3  2  3      3  5      3    3      3  4        2  5      3  3
psi  y  - psi  x  y  + psi  y  + psi  x y  - psi  y  - 3 psi  y  - psi  x

          3  2        3    2      3  3        2  2  2        2    3
     + psi  x  y + psi  x y  - psi  y  + 3 psi  x  y  - 3 psi  x y

            2  2          2    2        2  3          4        2  2
     + 3 psi  x  y - 8 psi  x y  + 5 psi  y  + 3 psi y  + 2 psi  x

            2            2  2          2              2          3          2
     - 4 psi  x y + 2 psi  y  - 3 psi x  y + 6 psi x y  - 3 psi y  - 3 psi x

                          2    3    2              2
     + 7 psi x y - 4 psi y  - y  + x  - 3 x y + 2 y  = 0



                             and in Maple  notation



psi^3*y^6-psi^3*x^2*y^3+psi^3*y^5+psi^3*x*y^3-psi^3*y^4-3*psi^2*y^5-psi^3*x^3+
psi^3*x^2*y+psi^3*x*y^2-psi^3*y^3+3*psi^2*x^2*y^2-3*psi^2*x*y^3+3*psi^2*x^2*y-8
*psi^2*x*y^2+5*psi^2*y^3+3*psi*y^4+2*psi^2*x^2-4*psi^2*x*y+2*psi^2*y^2-3*psi*x^
2*y+6*psi*x*y^2-3*psi*y^3-3*psi*x^2+7*psi*x*y-4*psi*y^2-y^3+x^2-3*x*y+2*y^2 = 0


              writing Psi(x,0)=g(x) as a Taylor series around x=0



                                   infinity
                                    -----
                                     \            n
                            g(x) =    )     a[n] x
                                     /
                                    -----
                                    n = 0



The coefficients, a[n], satisfy the folllowing linear recurrence equation wi\

    th polynomial coefficients of order, 2



        (3 n + 2) (3 n + 1) (3 n + 8) a(n)      9 a(n + 1)
   -3/4 ---------------------------------- - ----------------- + a(n + 2) = 0
            (3 n + 5) (n + 3) (n + 2)        (n + 3) (3 n + 5)



                             and in Maple notation



-3/4*(3*n+2)*(3*n+1)*(3*n+8)/(3*n+5)/(n+3)/(n+2)*a(n)-9/(n+3)/(3*n+5)*a(n+1)+a(
n+2) = 0


                       subject to the initial conditions



a(1) = 1, a(2) = 1


                    Finally, just for fun here is , a(1000)



a(1000) = 979126004443785412931142278503878513058808467397000810633549113420157\
8283149523637748958186355180833072004330547196414348651307900516137271566955606\
6750211464043783587913389858165786459115474543301980179084016196284587772296213\
2199044811127284633412163848291265268108776927426798214055878941986697359477539\
1532062721770795314545926837472474380661334765644319245879054765044780084512221\
7718615069863058443201120




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          This ends this paper that took, 0.794, seconds to generate