A305603 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].

1, 1, 8, 102, 1712, 34785, 819384, 21810124, 645122272, 20957720148, 741260263600, 28350052179438, 1165931175542064, 51320048879474206, 2407857124657086480, 119990501174741855400, 6330579163195128292800, 352584892981590315935084, 20675941712941698695206368, 1273517057922072215818491064, 82210136955409063394289646720

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..378

Formula

O.g.f. A(x) satisfies:

(1) [x^n] exp( n * Integral A(x)^3 dx ) * (n + 1 - A(x)) = 0 for n > 0.

(2) A(x) = 1 + x*A(x)^2*(A(x) + 3*x*A'(x))/(1 - x*A(x)^3)^2.

a(n) ~ c * 3^n * n^(4/3) * n!, where c = 0.1925904251831569484470022... - Vaclav Kotesovec, Oct 06 2020

Example

G.f.: A(x) = 1 + x + 8*x^2 + 102*x^3 + 1712*x^4 + 34785*x^5 + 819384*x^6 + 21810124*x^7 + 645122272*x^8 + 20957720148*x^9 + 741260263600*x^10 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].
RELATED SERIES.
A(x)^3 = 1 + 3*x + 27*x^2 + 355*x^3 + 5964*x^4 + 120021*x^5 + 2790794*x^6 + 73301427*x^7 + 2141393220*x^8 + 68800518492*x^9 + ...
1/(1 - x*A(x)^3) = 1 + x + 4*x^2 + 34*x^3 + 428*x^4 + 6957*x^5 + 136564*x^6 + 3115732*x^7 + 80640284*x^8 + 2328635572*x^9 + ...
A'(x)/A(x) = 1 + 15*x + 283*x^2 + 6343*x^3 + 162076*x^4 + 4614153*x^5 + 144287466*x^6 + 4908441479*x^7 + 180383821348*x^8 + 7122692545660*x^9 + ...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^3+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^3) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A305602, A305604, A209881, A305110.

Sequence in context: A001575 A366016 A328061 * A379690 A372334 A333985

Adjacent sequences: A305600 A305601 A305602 * A305604 A305605 A305606

Keyword

nonn

Author

Paul D. Hanna, Jun 05 2018

Status

approved