A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).

1, 1, 9, 159, 4051, 131688, 5132793, 231332589, 11778989157, 666865748751, 41494745678544, 2812781975630049, 206264308294757115, 16268935714201604701, 1373512281722006688063, 123601628009085259269819, 11812339040349301277253801, 1194940136210629914238593762

Offset: 1

Author

Paul D. Hanna, Mar 03 2023

Keywords

• nonn

Comments

Conjecture: a(n) == 1 (mod 3) iff n = (3^k - 1)/2 for k >= 0, otherwise a(n) == 0 (mod 3).

Examples

			G.f.: A(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^2) = x, where
A'(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + A361046(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^2) + (d^2/dx^2 x^6*A'(x)^4)/2! + (d^3/dx^3 x^9*A'(x)^6)/3! + (d^4/dx^4 x^12*A'(x)^8)/4! + (d^5/dx^5 x^15*A'(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^2 + (d/dx x^5*A'(x)^4)/2! + (d^2/dx^2 x^8*A'(x)^6)/3! + (d^3/dx^3 x^11*A'(x)^8)/4! + (d^4/dx^4 x^14*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(2*n))/n! + ... ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..301

Crossrefs

Cf. A361046, A229619, A360977, A360978.

Programs

PARI

{a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*(A')^2 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.

(1) A(x) = Series_Reversion(x - x^3*A'(x)^2).

(2) A(x) = x + A(x)^3 * A'(A(x))^2.

(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(2*n) / n! ).

(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(2*n) / n!, where A'(x) is the g.f. of A361046.

a(n) = A361046(n-1)/(2*n-1) for n >= 1.