A193302 E.g.f. satisfies: A(x) = x + Sum_{n>=1} x^n/n! * d^(n-1)/dx^(n-1) A_n(x), where A_n(x) denotes the n-th iteration of A(x), with A(0)=0.

1, 3, 36, 1800, 256320, 82293840, 54720315720, 74056548434400, 205691687907773760, 1188418644376456449600, 14442138586471988841926400, 371825786109253009955110351200, 20358746541959337235114022302070400, 2373555686236933938165235930052575843200

Offset

1,2

Links

Table of n, a(n) for n=1..14.

Example

E.g.f.: A(x) = x + 3*x^2/2! + 36*x^3/3! + 1800*x^4/4! + 256320*x^5/5! +...
where the e.g.f. satisfies:
A(x) = x + x*A(x) + x^2*(d/dx A(A(x)))/2! + x^3*(d^2/dx^2 A(A(A(x))))/3! + x^4*(d^3/dx^3 A(A(A(A(x)))))/4! +...

Mathematica

a[m_, cached_] := (c[0] = 0; If[cached > 0, Do[c[k] = c0[k], {k, 1, cached}]]; f[x_] = Series[ Sum[c[k] x^k , {k, 0, m}] , {x, 0, m}]; f[x_, n_] := D[Nest[f, u, n], {u, n - 1}] /. u -> x; cc = CoefficientList[ Series[ f[x] - x - Sum[x^n/n! f[x, n], {n, 1, m}], {x, 0, m}], x]; s = Solve[ Thread[cc == 0]]; Table[c0[k] = c[k] /. s[[1]] , {k, 1, m}] * Range[m]! ); a[5, 0]; a[10, 5]; Take[a[15, 10], 14] (* Jean-François Alcover, Sep 16 2011 *)

Prog

(PARI) /* n-th derivative of F: */
{Dx(F, n)=local(Dn=truncate(F)); for(i=1, n, Dn=deriv(Dn)); Dn}
/* n-th iteration of F: */
{ITERATE(F, n, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
{a(n)=local(A=x); for(i=1, n, A=x+sum(m=1, n, x^m/m!*Dx(ITERATE(A, m, n), m-1))+x*O(x^n)); n!*polcoeff(A, n)}

Crossrefs

Sequence in context: A233197 A300770 A272660 * A387812 A289315 A102579

Adjacent sequences: A193299 A193300 A193301 * A193303 A193304 A193305

Keyword

nonn

Author

Paul D. Hanna, Jul 21 2011

Status

approved