A220379 G.f. satisfies: A(x) = x + (1-x)*A(A(x))^2.
1, 1, 3, 16, 108, 836, 7136, 65708, 643522, 6638880, 71649726, 804833052, 9373104396, 112833093984, 1400634016520, 17894022203641, 234907670711601, 3164596264280695, 43700481254733535, 617995260359761384, 8942391804824517624, 132304132524112742604
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 3*x^3 + 16*x^4 + 108*x^5 + 836*x^6 + 7136*x^7 +... The g.f. satisfies the series: A(x) = x + (1-x)*A(x)^2 + (1-x)^2*d/dx A(x)^4/2! + (1-x)^3*d^2/dx^2 A(x)^6/3! + (1-x)^4*d^3/dx^3 A(x)^8/4! +... as well as the logarithmic series: log(A(x)/x) = (1-x)*A(x)^2/x + (1-x)^2*[d/dx A(x)^4/x]/2! + (1-x)^3*[d^2/dx^2 A(x)^6/x]/3! + (1-x)^4*[d^3/dx^3 A(x)^8/x]/4! +... Related expansions: A(A(x)) = x + 2*x^2 + 8*x^3 + 48*x^4 + 354*x^5 + 2958*x^6 + 27004*x^7 +... A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +... (A(x)-x)/(1-x) = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +... The series reversion of the g.f. A(x) equals: (x-A(x)^2)/(1-A(x)^2) = x - x^2 - x^3 - 6*x^4 - 34*x^5 - 234*x^6 - 1818*x^7 -... The series reversion of A(A(x)) equals: 1 - 1/((1+x)*(1-A(x)^2)) = x - 2*x^2 - 8*x^4 - 34*x^5 - 242*x^6 - 1852*x^7 -...
Crossrefs
Cf. A213591.
Programs
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PARI
{a(n)=local(A=x);for(i=1,n,A=x+(1-x)*subst(A,x,A+x*O(x^n))^2);polcoeff(A,n)} for(n=1,25,print1(a(n),", ")) -
PARI
{a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse((x - A^2)/(1-A^2+x*O(x^n)))); polcoeff(A, n))} for(n=1,25,print1(a(n),", ")) -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)} for(n=1,25,print1(a(n),", ")) -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)} for(n=1,25,print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) A(x) = x + Sum_{n>=1} (1-x)^n * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(2) A(x) = x*exp( Sum_{n>=1} (1-x)^n * d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
(3) A( (x - A(x)^2) / (1 - A(x)^2) ) = x.
(4) A(A( 1 - 1/((1+x)*(1-A(x)^2)) )) = x.
(5) A(A(x)) = sqrt( (A(x) - x) / (1 - x) ).