The linear recurrence operator annihilating the series of coefficients of th\
    e Onsager function, let's call it f(v)  is



      6       5        4       3       2            /d      \
(-13 v  + 70 v  - 251 v  + 16 v  + 69 v  + 2 v - 5) |-- f(v)|
                                                    \dv     /

                                                               / 2      \
                      5        4        3        2             |d       |
     - v (v - 1) (55 v  - 309 v  + 328 v  + 220 v  - 87 v + 1) |--- f(v)|
                                                               |  2     |
                                                               \dv      /

                                                         / 3      \
        2      4        3       2                      2 |d       |
     - v  (33 v  - 172 v  - 10 v  + 116 v - 15) (v - 1)  |--- f(v)|
                                                         |  3     |
                                                         \dv      /

                                            / 4      \
          3           2                   3 |d       |
     - 4 v  (v + 1) (v  - 6 v + 1) (v - 1)  |--- f(v)| = 0
                                            |  4     |
                                            \dv      /



                             and in Maple notation



(-13*v^6+70*v^5-251*v^4+16*v^3+69*v^2+2*v-5)*diff(f(v),v)-v*(v-1)*(55*v^5-309*v
^4+328*v^3+220*v^2-87*v+1)*diff(diff(f(v),v),v)-v^2*(33*v^4-172*v^3-10*v^2+116*
v-15)*(v-1)^2*diff(diff(diff(f(v),v),v),v)-4*v^3*(v+1)*(v^2-6*v+1)*(v-1)^3*diff
(diff(diff(diff(f(v),v),v),v),v) = 0


                          the differential operator is



(-13*v^6+70*v^5-251*v^4+16*v^3+69*v^2+2*v-5)*Dv-v*(v-1)*(55*v^5-309*v^4+328*v^3
+220*v^2-87*v+1)*Dv^2-v^2*(33*v^4-172*v^3-10*v^2+116*v-15)*(v-1)^2*Dv^3-4*v^3*(
v+1)*(v^2-6*v+1)*(v-1)^3*Dv^4