Find Ge[z]
:
r1 = p[z] -> 2 z + E^t[z] - 1 - t[z]
r2 = t[z] -> 2*(z + E^p[z] - E^(p[z]/2)) - p[z]
Ge[z_] = p[z] + t[z] - 2 z /. r1 /. r2 // FullSimplify
(* -1 + E^(2 (-E^((p[z]/2)) + E^p[z] + z) - p[z]) *)
SeriesCoefficient
produces derivatives of p
at z == 0
. Therefore we have to solve for the p[0]
, p'[0]
, p''[0]
...
der1[i_] := Derivative[i][p][0] == Derivative[i][2 # + E^t[#] - 1 - t[#] &][0]
der2[i_] :=
Derivative[i][t][0] == Derivative[i][2*(# + E^p[#] - E^(p[#]/2)) - p[#] &][0]
sol[0] = First @
Solve[{der1[0], der2[0]}, {Derivative[0][p][0],
Derivative[0][t][0]}, Reals]
(* {p[0] -> 0, t[0] -> 0} *)
Lower derivatives are needed to solve for higher ones:
sol[j_ /; j > 0] := sol[j]=
First@Solve[{der1[j], der2[j]} //.
Flatten[Table[sol[k], {k, 0, j - 1}]], {Derivative[j][p][0],
Derivative[j][t][0]}, Reals]
Inserting these found derivatives of p
into the series coefficients gives the desired A
numbers.
A[n_] := n! SeriesCoefficient[Ge[z], {z, 0, n}] //.
Flatten[Table[sol[i], {i, 0, n}]]
Table[A[i], {i, 1, 10}]
(* {2, 10, 94, 1466, 31814, 887650, 30259198, 1218864842, 56644903958,
2983300619410} *)