A306404 E.g.f. A(x) satisfies: A(x) = (1 + Integral A(x) dx) * (1 + Integral A(x)^3 dx).

1, 2, 10, 88, 1088, 17296, 336160, 7722944, 204747904, 6152445568, 206635255040, 7670855683072, 311892151155712, 13784371218151424, 657962006198824960, 33732998333486350336, 1848747736087419723776, 107859057160535649206272, 6674104727394168140922880, 436582063054208216587501568, 30102600290916367728363962368, 2182043529056049327839246811136

Offset

0,2

Comments

Compare: G(x) = (1 + Integral G(x) dx)^2 holds when G(x) = 1/(1 - x)^2.

Compare: G(x) = (1 + Integral G(x)^2 dx)^2 holds when G(x) = 1/(1 - 3*x)^(2/3), the e.g.f. of the triple factorials product_{k=0..n-1} (3*k+2).

Compare: G(x) = (1 + Integral G(x)^m dx)^2 holds when G(x) = 1/(1 - (2*m-1)*x)^(2/(2*m-1)) = Sum_{n>=0} x^n/n! * product_{k=0..n-1} ((2*m-1)*k + 2).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f. A(x) satisfies the following relations.

(1) A(x) = (1 + Integral A(x) dx) * (1 + Integral A(x)^3 dx).

(2) A'(x) = A(x) * (1 + Integral A(x)^3 dx) + A(x)^3 * (1 + Integral A(x) dx).

(3) log(A(x)) = Integral [ A(x)/(1 + Integral A(x) dx) + A(x)^3/(1 + Integral A(x)^3 dx) ] dx.

(4a) log(1 + Integral A(x) dx) = Integral (1 + Integral A(x)^3 dx) dx.

(4b) log(1 + Integral A(x)^3 dx) = Integral A(x)^2*(1 + Integral A(x) dx) dx.

a(n) ~ c * d^n * n^n, where d = 9*exp(-1) / (8*log(2) - 3) = 1.30085820842247653985772994360460264422544953483565... and c = 1.4925156370342369979236718531290597194906869115... - Vaclav Kotesovec, Aug 11 2021, updated Apr 19 2024

Example

E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 88*x^3/3! + 1088*x^4/4! + 17296*x^5/5! + 336160*x^6/6! + 7722944*x^7/7! + 204747904*x^8/8! + 6152445568*x^9/9! + ...
RELATED SERIES.
A(x)^3 = 1 + 6*x + 54*x^2/2! + 672*x^3/3! + 10728*x^4/4! + 209088*x^5/5! + 4812912*x^6/6! + 127780416*x^7/7! + 3843863424*x^8/8! + ...
log(A(x)) = 2*x + 6*x^2/2! + 44*x^3/3! + 468*x^4/4! + 6624*x^5/5! + 117168*x^6/6! + 2486592*x^7/7! + 61560864*x^8/8! + 1741698432*x^9/9! + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + intformal( A )) * (1 + intformal( A^3 +x*O(x^n))) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A322738.

Sequence in context: A185388 A245009 A354240 * A111811 A186448 A144002

Adjacent sequences: A306401 A306402 A306403 * A306405 A306406 A306407

Keyword

nonn,changed

Author

Paul D. Hanna, Mar 08 2019

Status

approved