A361307 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^4).

1, 1, 15, 462, 20719, 1187628, 81575478, 6470236914, 578865763791, 57491440616067, 6266161502595672, 743009082083639748, 95191896469891628934, 13103364445591714775407, 1928820020328686200102278, 302383969785427961077318020, 50307405653295945234562827135

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

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)^4).

(2) A(x) = x + A(x)^3 * A'(A(x))^4.

(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(4*n) / n! ).

(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(4*n) / n! is the g.f. of A361537.

(5) a(n) = A361537(n-1)/(2*n-1) for n >= 1.

Example

G.f.: A(x) = x + x^3 + 15*x^5 + 462*x^7 + 20719*x^9 + 1187628*x^11 + 81575478*x^13 + 6470236914*x^15 + 578865763791*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^4) = x, where
A'(x) = 1 + 3*x^2 + 75*x^4 + 3234*x^6 + 186471*x^8 + 13063908*x^10 + 1060481214*x^12 + 97053553710*x^14 + ... + A361537(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^4) + (d^2/dx^2 x^6*A'(x)^8)/2! + (d^3/dx^3 x^9*A'(x)^12)/3! + (d^4/dx^4 x^12*A'(x)^16)/4! + (d^5/dx^5 x^15*A'(x)^20)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(4*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^4 + (d/dx x^5*A'(x)^8)/2! + (d^2/dx^2 x^8*A'(x)^12)/3! + (d^3/dx^3 x^11*A'(x)^16)/4! + (d^4/dx^4 x^14*A'(x)^20)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(4*n))/n! + ... ).

Prog

(PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^4 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A361537.

Cf. A229619, A360976, A360977, A360978, A361302, A361308, A361309, A361310, A361311.

Sequence in context: A306675 A219561 A209488 * A320097 A177080 A306609

Adjacent sequences: A361304 A361305 A361306 * A361308 A361309 A361310

Keyword

nonn

Author

Paul D. Hanna, Mar 17 2023

Status

approved