This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A210949 #38 May 13 2025 12:02:00
%S A210949 1,1,4,29,309,4383,78121,1684706,42801222,1255919755,41918624013,
%T A210949 1572257236114,65619165625383,3022617826829288,152615633802149416,
%U A210949 8397224009015443509,500957609480739613321,32261529179806961067634,2234133327582388824135291
%N A210949 E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(A(x))).
%C A210949 An unsigned version of A067146.
%C A210949 Equals row sums of triangle A277410.
%C A210949 Is this a duplicate of A014622, which is related to f(f(x))f'(x)=-1 ? - _R. J. Mathar_, May 13 2025
%H A210949 Vaclav Kotesovec, <a href="/A210949/b210949.txt">Table of n, a(n) for n = 1..180</a>
%F A210949 E.g.f. A(x) satisfies:
%F A210949 (1) A(x) = Series_Reversion( Integral 1 - A(x) dx ).
%F A210949 (2) A''(x) = 1 / ( (1 - A(A(x)))^3 * (1 - A(A(A(x)))) ).
%F A210949 Let G(x) = Integral A(x) dx with G(0)=0, then the e.g.f. A(x) satisfies:
%F A210949 (3) A(x) = x + G(A(x)) or, equivalently, A(x - G(x)) = x.
%F A210949 (4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) G(x)^n / n!.
%F A210949 (5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) G(x)^n/(n!*x) ).
%F A210949 a(n) = Sum_{k=0..n-1} A277410(n,k).
%e A210949 E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 29*x^4/4! + 309*x^5/5! + 4383*x^6/6! +...
%e A210949 Let G(x) = Integral A(x) dx, then A(x) = x + G(A(x)) where
%e A210949 G(x) = x^2/2! + x^3/3! + 4*x^4/4! + 29*x^5/5! + 309*x^6/6! + 4383*x^7/7! +...
%e A210949 Also,
%e A210949 A(x) = x + G(x) + d/dx G(x)^2/2! + d^2/dx^2 G(x)^3/3! + d^3/dx^3 G(x)^4/4! +...
%e A210949 log(A(x)/x) = G(x)/x + d/dx G(x)^2/(2!*x) + d^2/dx^2 G(x)^3/(3!*x) + d^3/dx^3 G(x)^4/(4!*x) +...
%e A210949 By definition, A'(x) = 1/(1 - A(A(x))), where
%e A210949 A(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 101*x^4/4! + 1313*x^5/5! + 22235*x^6/6! + 466356*x^7/7! + 11710760*x^8/8! +...
%t A210949 m = 20; A[_] = 0;
%t A210949 Do[A[x_] = InverseSeries[Integrate[1 - A[x], x] + O[x]^m], {m}];
%t A210949 CoefficientList[A[x], x] * Range[0, m - 1]! // Rest (* _Jean-François Alcover_, Sep 30 2019 *)
%o A210949 (PARI) {a(n)=local(A=x,G);for(i=1,n,G=intformal(A+x*O(x^n));A=serreverse(x-G));n!*polcoeff(A, n)}
%o A210949 (PARI) {a(n)=local(A=x,G);for(i=1,n,G=intformal(A+x*O(x^n));A=x+subst(G,x,A+x*O(x^n))); n!*polcoeff(A, n)}
%o A210949 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A210949 {a(n)=local(A=x,G);for(i=0,n,G=intformal(A+x*O(x^n)); A=x+sum(m=1, n, Dx(m-1, G^m/m!))); n!*polcoeff(A, n)}
%o A210949 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A210949 {a(n)=local(A=x,G);for(i=0,n,G=intformal(A+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, G^m/x/m!)+x*O(x^n)))); n!*polcoeff(A, n)}
%o A210949 for(n=1, 25, print1(a(n), ", "))
%o A210949 (PARI) {a(n) = local(A=x); for(i=1,n, A = serreverse(intformal(1-A +x*O(x^n)))); n!*polcoeff(A,n)}
%o A210949 for(n=1, 25, print1(a(n), ", "))
%Y A210949 Cf. A277410, A067146, A279843, A279844, A279845, A280570, A280571, A280572, A280573, A280574, A280575.
%K A210949 nonn
%O A210949 1,3
%A A210949 _Paul D. Hanna_, Jul 22 2012