A372577 Expansion of g.f. A(x) satisfying A(x)^2 = A(A( x*A(x) + x*A(x)^2 )).

1, 1, 3, 9, 33, 125, 497, 2033, 8523, 36405, 157889, 693393, 3077321, 13780073, 62183839, 282499233, 1290970465, 5930362445, 27369471425, 126842296553, 590061195147, 2754310145213, 12896723586205, 60559396163137, 285113800165521, 1345551648937105, 6364274750580531

Offset

1,3

Comments

Compare to C(x)^2 = C( x*C(x) + x*C(x)^2 ) when C(x) = x + C(x)^2 is the Catalan function (A000108).

All terms appear to be odd.

Conjecture: a(6*n - 3) == 3 (mod 4) and a(6*n - k) == 1 (mod 4) when k = {0,1,2,4,5} for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x)^2 = A(A( x*A(x)*(1 + A(x)) )).

(2) A(x)^4 = A(A( A(x)^2*(1 + A(x)^2) * A(x*A(x)*(1 + A(x))) )).

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 33*x^5 + 125*x^6 + 497*x^7 + 2033*x^8 + 8523*x^9 + 36405*x^10 + 157889*x^11 + 693393*x^12 + 3077321*x^13 + ...
satisfies A(x)^2 = A(A( x*A(x) + x*A(x)^2 )).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 24*x^5 + 93*x^6 + 370*x^7 + 1523*x^8 + 6404*x^9 + 27433*x^10 + 119250*x^11 + 524747*x^12 + 2332836*x^13 + ...
x*A(x) + x*A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 57*x^6 + 218*x^7 + 867*x^8 + 3556*x^9 + 14927*x^10 + 63838*x^11 + 277139*x^12 + ...
A( x*A(x) + x*A(x)^2 ) = x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 74*x^6 + 288*x^7 + 1160*x^8 + 4800*x^9 + 20282*x^10 + 87176*x^11 + 379936*x^12 + ...
Let B(x) = A(A(x)) then
B(x) = x + 2*x^2 + 8*x^3 + 34*x^4 + 162*x^5 + 808*x^6 + 4190*x^7 + 22334*x^8 + 121708*x^9 + 675142*x^10 + 3800826*x^11 + 21664832*x^12 + ...
satisfies B(x)^2 = B( A(x)*B(x)*(1 + B(x)) ).
Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^3 + x^4 - x^5 + x^6 + 5*x^7 - 5*x^8 - 5*x^9 + 5*x^10 - x^11 + x^12 + 17*x^13 - 17*x^14 - 69*x^15 + 69*x^16 + 99*x^17 - 99*x^18 + ...
satisfies R(R(x^2)) = x*(1+x)*R(x).
SPECIFIC VALUES.
A(x) appears to diverge at x = 1/5.
A(t) = 1/3 at t = 0.1961428756801475570678466959574244574032970008463260646...
A(1/6) = 0.2283714063489089174991793877683763254200090211944062418...
A(1/7) = 0.1803022958562088529850234174666856970193109686048442373...
A(1/8) = 0.1506624579090123122038145735552538129097334092730102352...
A(1/10) = 0.1144400257041317214982655649694510141147195315483064558...

Prog

(PARI) {a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A);
A[#A] = polcoeff( subst(F, x, subst(F, x, x*F + x*F^2 )) - F^2, #A+1) ); H=F; A[n]}
for(n=1, 35, print1(a(n), ", "))

Crossrefs

Sequence in context: A307454 A219261 A182530 * A049182 A372530 A049168

Adjacent sequences: A372574 A372575 A372576 * A372578 A372580 A372581

Keyword

nonn

Author

Paul D. Hanna, May 30 2024

Status

approved