A361302 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^3).

1, 1, 12, 291, 10243, 460632, 24830853, 1546531419, 108716955930, 8489321379453, 727903248520260, 67935651633100242, 6853940772480079902, 743261410711529857459, 86224073603509482578211, 10656471864208782754351131, 1398062659621217619155428209

Offset

1,3

Links

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

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)^3).

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

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

(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(3*n) / n! is the g.f. of A361536.

(5) a(n) = A361536(n-1)/(2*n-1) for n >= 1.

Example

G.f.: A(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + 1546531419*x^15 + 108716955930*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^3) = x, where
A'(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + A361536(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^3) + (d^2/dx^2 x^6*A'(x)^6)/2! + (d^3/dx^3 x^9*A'(x)^9)/3! + (d^4/dx^4 x^12*A'(x)^12)/4! + (d^5/dx^5 x^15*A'(x)^15)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(3*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^3 + (d/dx x^5*A'(x)^6)/2! + (d^2/dx^2 x^8*A'(x)^9)/3! + (d^3/dx^3 x^11*A'(x)^12)/4! + (d^4/dx^4 x^14*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(3*n))/n! + ... ).

Prog

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

Crossrefs

Cf. A361536.

Cf. A229619, A360976, A360977, A360978, A361307, A361308, A361309, A361310, A361311.

Sequence in context: A041267 A041264 A220307 * A361309 A223011 A245915

Adjacent sequences: A361299 A361300 A361301 * A361303 A361304 A361305

Keyword

nonn

Author

Paul D. Hanna, Mar 17 2023

Status

approved