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A001028 E.g.f. satisfies A'(x) = 1 + A(A(x)), A(0)=0. 10
1, 1, 2, 7, 37, 269, 2535, 29738, 421790, 7076459, 138061343, 3089950076, 78454715107, 2238947459974, 71253947372202, 2511742808382105, 97495087989736907, 4145502184671892500, 192200099033324115855, 9676409879981926733908, 527029533717566423156698 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

Alois P. Heinz and 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.

FORMULA

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

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-Fran├žois Alcover, Dec 07 2011, updated Jan 10 2018 *)

PROG

(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

CROSSREFS

Cf. A030266, A035049.

Sequence in context: A036247 A083659 A107877 * A116481 A102743 A195068

Adjacent sequences:  A001025 A001026 A001027 * A001029 A001030 A001031

KEYWORD

nonn,eigen,nice

AUTHOR

Peter J. Cameron

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

STATUS

approved

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Last modified June 25 17:47 EDT 2019. Contains 324353 sequences. (Running on oeis4.)