A001028 - OEIS

%I #68 Oct 27 2023 18:03:31

%S 1,1,2,7,37,269,2535,29738,421790,7076459,138061343,3089950076,

%T 78454715107,2238947459974,71253947372202,2511742808382105,

%U 97495087989736907,4145502184671892500,192200099033324115855,9676409879981926733908,527029533717566423156698

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

%C The e.g.f. is diverging (see the Math Overflow link). - _Pietro Majer_, Jan 29 2017

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

%H Vaclav Kotesovec, <a href="/A001028/b001028.txt">Table of n, a(n) for n = 1..320</a> (first 100 terms from Alois P. Heinz)

%H P. J. Cameron, <a href="http://dx.doi.org/10.1016/0024-3795(95)00352-R">Sequence operators from groups</a>, Linear Alg. Applic., 226-228 (1995), 109-113.

%H Math Overflow, <a href="http://mathoverflow.net/questions/258611/f-ef-1-again/258991#258991">f' = exp(f^(-1)), again</a>, January 2017.

%F E.g.f. satisfies: A(x) = Series_Reversion( Integral 1/(1 + A(x)) dx ). - _Paul D. Hanna_, Jun 27 2015

%p 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

%t terms = 21; A[_] = 0; Do[A[x_] = x + Integrate[A[A[x]], x] + O[x]^(n+1) // Normal, {n, terms}];

%t Rest[CoefficientList[A[x], x]]*Range[terms]! (* _Jean-François Alcover_, Dec 07 2011, updated Jan 10 2018 *)

%o (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 */

%o (PARI) {a(n) = my(A=x); for(i=1,n, A = serreverse(intformal(1/(1+A) +x*O(x^n)))); n!*polcoeff(A,n)}

%o for(n=1,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Jun 27 2015

%Y Cf. A030266, A035049.

%K nonn,eigen,nice

%O 1,3

%A _Peter J. Cameron_

%E More terms from _Christian G. Bower_, Oct 15 1998

%E Corrected by _Alois P. Heinz_, Aug 23 2008

%E Two more terms from _Sean A. Irvine_, Feb 22 2012