A210949 - OEIS

%I #34 Sep 30 2019 08:51:30

%S 1,1,4,29,309,4383,78121,1684706,42801222,1255919755,41918624013,

%T 1572257236114,65619165625383,3022617826829288,152615633802149416,

%U 8397224009015443509,500957609480739613321,32261529179806961067634,2234133327582388824135291

%N E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(A(x))).

%C An unsigned version of A067146.

%C Equals row sums of triangle A277410.

%H Vaclav Kotesovec, <a href="/A210949/b210949.txt">Table of n, a(n) for n = 1..180</a>

%F E.g.f. A(x) satisfies:

%F (1) A(x) = Series_Reversion( Integral 1 - A(x) dx ).

%F (2) A''(x) = 1 / ( (1 - A(A(x)))^3 * (1 - A(A(A(x)))) ).

%F Let G(x) = Integral A(x) dx with G(0)=0, then the e.g.f. A(x) satisfies:

%F (3) A(x) = x + G(A(x)) or, equivalently, A(x - G(x)) = x.

%F (4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) G(x)^n / n!.

%F (5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) G(x)^n/(n!*x) ).

%F a(n) = Sum_{k=0..n-1} A277410(n,k).

%e 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 Let G(x) = Integral A(x) dx, then A(x) = x + G(A(x)) where

%e 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 Also,

%e 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 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 By definition, A'(x) = 1/(1 - A(A(x))), where

%e 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 m = 20; A[_] = 0;

%t Do[A[x_] = InverseSeries[Integrate[1 - A[x], x] + O[x]^m], {m}];

%t CoefficientList[A[x], x] * Range[0, m - 1]! // Rest (* _Jean-François Alcover_, Sep 30 2019 *)

%o (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 (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 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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 for(n=1, 25, print1(a(n), ", "))

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

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A277410, A067146, A279843, A279844, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 22 2012