A212910 - OEIS

%I #14 Aug 24 2017 10:09:33

%S 1,1,1,4,11,35,125,445,1699,6668,26935,112111,476674,2072146,9182141,

%T 41406119,189830984,883549848,4171598085,19962224926,96746007976,

%U 474586282085,2355104582435,11817111373152,59928222117495,307045555880793,1588825668984517

%N G.f. satisfies: A(x) = x^2 - x + Series_Reversion(x - x*A(x)).

%C Compare the g.f. to a g.f. G(x) of A088714 (offset 1), which satisfies:

%C G(x) = Series_Reversion(x - x*G(x)),

%C G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*G(x)^n/n!, and

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

%H Vaclav Kotesovec, <a href="/A212910/b212910.txt">Table of n, a(n) for n = 2..375</a>

%F G.f. A(x) also satisfies:

%F (1) A(x) = x^2 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.

%F (2) A(x) = x^2 - x + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).

%e G.f.: A(x) = x^2 + x^3 + x^4 + 4*x^5 + 11*x^6 + 35*x^7 + 125*x^8 +...

%e The series reversion of x - x*A(x) begins:

%e x + x^3 + x^4 + 4*x^5 + 11*x^6 + 35*x^7 + 125*x^8 + 445*x^9 +...

%e which equals x - x^2 + A(x).

%e The g.f. satisfies:

%e A(x) = x^2 + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...

%e log(1-x + A(x)/x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...

%e Related expansions:

%e d/dx x^2*A(x)^2/2! = 3*x^5 + 7*x^6 + 12*x^7 + 45*x^8 + 155*x^9 +...

%e d^2/dx^2 x^3*A(x)^3/3! = 12*x^7 + 45*x^8 + 110*x^9 + 418*x^10 +...

%e d^3/dx^3 x^4*A(x)^4/4! = 55*x^9 + 286*x^10 + 910*x^11 + 3640*x^12 +...

%e d^4/dx^4 x^5*A(x)^5/5! = 273*x^11 + 1820*x^12 + 7140*x^13 +...

%e ...

%e d^(n-1)/dx^(n-1) x^n*A(x)^n/n! = A001764(n)*x^(2*n+1) +...

%o (PARI) {a(n)=local(A=x^2);for(i=1,n,A=x^2-x+serreverse(x-x*A +x*O(x^n)));polcoeff(A,n)}

%o for(n=2,35,print1(a(n),", "))

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

%o {a(n)=local(A=x^2);for(i=1,n,A=x^2+sum(m=1,n,Dx(m-1,x^m*A^m/m!)+x*O(x^n)));polcoeff(A,n)}

%o for(n=2,35,print1(a(n),", "))

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

%o {a(n)=local(A=x^2);for(i=1,n,A=x^2-x+x*exp(sum(m=1,n,Dx(m-1,x^(m-1)*A^m/m!)+x*O(x^n))));polcoeff(A,n)}

%o for(n=2,35,print1(a(n),", "))

%Y Cf. A088714, A212919, A212922, A212923.

%K nonn

%O 2,4

%A _Paul D. Hanna_, May 30 2012