A360977 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^2).

1, 1, 6, 65, 978, 18354, 406396, 10274553, 290411050, 9045087486, 307195803380, 11287142857810, 445880114920868, 18841715641048244, 848104833677384856, 40515633401768290017, 2047629885363936119690, 109168898899192961652766, 6124201250273921324876116

Offset

1,3

Comments

a(n) = A360974(n-1)/n for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

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

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

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

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

(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A'(x)^(2*n) / n!.

Example

G.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 978*x^5 + 18354*x^6 + 406396*x^7 + 10274553*x^8 + 290411050*x^9 + ...
By definition, A(x - x^2*A'(x)^2) = x, where
A'(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + ... + A360974(n)*x^n + ...
Also,
A'(x) = 1 + (d/dx x^2*A'(x)^2) + (d^2/dx^2 x^4*A'(x)^4)/2! + (d^3/dx^3 x^6*A'(x)^6)/3! + (d^4/dx^4 x^8*A'(x)^8)/4! + (d^5/dx^5 x^10*A'(x)^10)/5! + ... + (d^n/dx^n x^(2*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x*A'(x)^2 + (d/dx x^3*A'(x)^4)/2! + (d^2/dx^2 x^5*A'(x)^6)/3! + (d^3/dx^3 x^7*A'(x)^8)/4! + (d^4/dx^4 x^9*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A'(x)^(2*n))/n! + ... ).

Prog

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

Crossrefs

Cf. A360974, A229619, A360976, A360978.

Sequence in context: A297516 A099343 A189509 * A243698 A217899 A349524

Adjacent sequences: A360974 A360975 A360976 * A360978 A360979 A360980

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2023

Status

approved