A396845 - OEIS

%I #7 Jul 03 2026 11:36:05

%S 1,5,45,575,8005,118925,1848175,29683375,488787945,8207945825,

%T 140023324375,2419926703375,42278635863875,745496061139125,

%U 13249931378913125,237123418577294375,4269329970987934055,77279684433882711925,1405520392608578792025,25672052588664668494375,470709407862301830257925

%N G.f. A(x) satisfies A( x*A(x) - 5*x*A(x)^2 ) = x^2.

%H Paul D. Hanna, <a href="/A396845/b396845.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with B(x) where B(A(x)) = x, satisfies the following formulas.

%F (1) B(x) = B(B(x)^2) / (x - 5*x^2).

%F (2) B(x^2) = x*A(x) - 5*x*A(x)^2.

%F (3) A(x) - 5*A(x)^2 is an odd function.

%F (4) A(x) = (1 - sqrt(1 - 20*B(x^2)/x)) / 10.

%F (5) A(x) = (1/5)*C( 5*B(x^2)/x ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.

%F The radius of convergence r satisfies r^2 = A(r/20), A(r) = 1/10, and A(-r) = -(sqrt(2) - 1)/10 where r = 0.050648013557126794001613224411704207390680989556692787...

%e G.f.: A(x) = x + 5*x^2 + 45*x^3 + 575*x^4 + 8005*x^5 + 118925*x^6 + 1848175*x^7 + 29683375*x^8 + 488787945*x^9 + 8207945825*x^10 + ...

%e such that A( x*A(x) - 5*x*A(x)^2 ) = x^2

%e where

%e A(x)^2 = x^2 + 10*x^3 + 115*x^4 + 1600*x^5 + 23785*x^6 + 369650*x^7 + 5936675*x^8 + 97757500*x^9 + 1641589165*x^10 + ...

%e and

%e A(x) - 5*A(x)^2 = x - 5*x^3 + 5*x^5 - 75*x^7 + 445*x^9 - 2375*x^11 + 18875*x^13 - 135625*x^15 + 971555*x^17 - 7626225*x^19 + ...

%e which is an odd function.

%e Compare with B(x), the series reversion of A(x), A(B(x)) = x:

%e B(x) = x - 5*x^2 + 5*x^3 - 75*x^4 + 445*x^5 - 2375*x^6 + 18875*x^7 - 135625*x^8 + 971555*x^9 - 7626225*x^10 + ...

%e which satisfies: B(B(x)^2) = (x - 5*x^2)*B(x).

%e SPECIFIC VALUES.

%e A(1/20) = 0.0888331272418844744627229655682642865491084541976232...

%e A(1/21) = 0.0758352831248204153051711824829869843447396211754731...

%e A(1/22) = 0.0683327114842221656011473194179884705760694623166237...

%e where 1/22^2 = A( (1/22)*(A(1/22) - 5*A(1/22)^2) ).

%o (PARI) {a(n,k=5) = my(A=x); for(i=1, #binary(n), A = (1 - sqrt(1 - 4*k*subst(serreverse(A +x*O(x^n)), x, x^2)/x) )/(2*k) ); polcoeff(GF=A, n)}

%o {upto(n) = a(n,5); Vec(GF)}

%o upto(25)

%Y Cf. A265940, A271958, A396843, A396844.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 03 2026