A001028 E.g.f. satisfies A'(x) = 1 + A(A(x)), A(0)=0.

1, 1, 2, 7, 37, 269, 2535, 29738, 421790, 7076459, 138061343, 3089950076, 78454715107, 2238947459974, 71253947372202, 2511742808382105, 97495087989736907, 4145502184671892500, 192200099033324115855, 9676409879981926733908, 527029533717566423156698

Offset: 1

Author

Peter J. Cameron

Keywords

• nonn
• eigen
• nice

Comments

The e.g.f. is diverging (see the Math Overflow link). - Pietro Majer, Jan 29 2017

References

• This functional equation (for f(x)=1+A(x-1)) was the subject of problem B5 of the 2010 Putnam exam.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..320 (first 100 terms from Alois P. Heinz)
• P. J. Cameron, Sequence operators from groups, Linear Alg. Applic., 226-228 (1995), 109-113.
• Math Overflow, f' = exp(f^(-1)), again, January 2017.

Crossrefs

Cf. A030266, A035049.

Programs

Maple

A:= proc(n) option remember; local T; if n=0 then 0 else T:= A(n-1); unapply(convert(series(Int(1+T(T(x)), x), x, n+1), polynom), x) fi end: a:= n-> coeff(A(n)(x), x, n)*n!: seq(a(n), n=1..22); # Alois P. Heinz, Aug 23 2008

Mathematica

terms = 21; A[] = 0; Do[A[x] = x + Integrate[A[A[x]], x] + O[x]^(n+1) // Normal, {n, terms}];
Rest[CoefficientList[A[x], x]]*Range[terms]! (* Jean-François Alcover, Dec 07 2011, updated Jan 10 2018 *)

Maxima

Co(n,k,a):= if k=1 then a(n) else sum(a(i+1)*Co(n-i-1,k-1,a), i,0,n-k); a(n):= if n=1 then 1 else (1/n)*sum(Co(n-1,k,a)*a(k),k,1,n-1); makelist(n!*a(n),n,1,7); /* Vladimir Kruchinin, Jun 30 2011 */

PARI

{a(n) = my(A=x); for(i=1,n, A = serreverse(intformal(1/(1+A) +x*O(x^n)))); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", ")) \\ Paul D. Hanna, Jun 27 2015

Formula

E.g.f. satisfies: A(x) = Series_Reversion( Integral 1/(1 + A(x)) dx ). - Paul D. Hanna, Jun 27 2015

Extensions

More terms from Christian G. Bower, Oct 15 1998

Corrected by Alois P. Heinz, Aug 23 2008

Two more terms from Sean A. Irvine, Feb 22 2012