A360978 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^3).

1, 1, 8, 119, 2476, 64370, 1974468, 69109563, 2702001936, 116298977966, 5453395749960, 276403464191890, 15049886389916756, 875933263547340216, 54268470230312961400, 3566244291096016078419, 247800396100716098128236, 18155541676448293842945990

Offset

1,3

Comments

a(n) = A360975(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)^3).

(2) A(x) = x + A(x)^2 * A’(A(x))^3.

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

(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A'(x)^(3*n) / n!.

Example

G.f.: A(x) = x + x^2 + 8*x^3 + 119*x^4 + 2476*x^5 + 64370*x^6 + 1974468*x^7 + 69109563*x^8 + 2702001936*x^9 + ...
By definition, A(x - x^2*A'(x)^3) = x, where
A'(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + ... + A360975(n)*x^n + ...
Also,
A'(x) = 1 + (d/dx x^2*A'(x)^3) + (d^2/dx^2 x^4*A'(x)^6)/2! + (d^3/dx^3 x^6*A'(x)^9)/3! + (d^4/dx^4 x^8*A'(x)^12)/4! + (d^5/dx^5 x^10*A'(x)^15)/5! + ... + (d^n/dx^n x^(2*n)*A'(x)^(3*n))/n! + ...
Further,
A(x) = x * exp( x*A'(x)^3 + (d/dx x^3*A'(x)^6)/2! + (d^2/dx^2 x^5*A'(x)^9)/3! + (d^3/dx^3 x^7*A'(x)^12)/4! + (d^4/dx^4 x^9*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A'(x)^(3*n))/n! + ... ).

Prog

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

Crossrefs

Cf. A360975, A229619, A360976, A360977.

Sequence in context: A318009 A368911 A228752 * A116008 A086302 A053129

Adjacent sequences: A360975 A360976 A360977 * A360979 A360980 A360981

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2023

Status

approved