A361308 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)).

1, 1, 8, 122, 2676, 75197, 2548336, 100461956, 4500071172, 225305924896, 12456434569184, 753380353835754, 49473301917640864, 3505613955205438686, 266627715169575108168, 21667902182055638829520, 1873978995774161192935320, 171874439346918445003163152

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^(3*n-2) may be defined by the following.

(1) A(x) = Series_Reversion(x - x^4*A'(x)).

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

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

(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^n / n! is the g.f. of A361541.

(5) a(n) = A361541(n-1)/(3*n-2) for n >= 1.

Example

G.f.: A(x) = x + x^4 + 8*x^7 + 122*x^10 + 2676*x^13 + 75197*x^16 + 2548336*x^19 + 100461956*x^22 + 4500071172*x^25 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)) = x, where
A'(x) = 1 + 4*x^3 + 56*x^6 + 1220*x^9 + 34788*x^12 + 1203152*x^15 + 48418384*x^18 + 2210163032*x^21 + ... + A361541(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)) + (d^2/dx^2 x^8*A'(x)^2)/2! + (d^3/dx^3 x^12*A'(x)^3)/3! + (d^4/dx^4 x^16*A'(x)^4)/4! + (d^5/dx^5 x^20*A'(x)^5/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^3*A'(x) + (d/dx x^7*A'(x)^2)/2! + (d^2/dx^2 x^11*A'(x)^3)/3! + (d^3/dx^3 x^15*A'(x)^4)/4! + (d^4/dx^4 x^19*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^n)/n! + ... ).

Prog

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

Crossrefs

Cf. A361541.

Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361309, A361310, A361311.

Sequence in context: A196972 A197559 A222498 * A359355 A349525 A240477

Adjacent sequences: A361305 A361306 A361307 * A361309 A361310 A361311

Keyword

nonn

Author

Paul D. Hanna, Mar 17 2023

Status

approved