A306404 - OEIS

%I #10 Apr 19 2024 03:05:53

%S 1,2,10,88,1088,17296,336160,7722944,204747904,6152445568,

%T 206635255040,7670855683072,311892151155712,13784371218151424,

%U 657962006198824960,33732998333486350336,1848747736087419723776,107859057160535649206272,6674104727394168140922880,436582063054208216587501568,30102600290916367728363962368,2182043529056049327839246811136

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

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

%C 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).

%C 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).

%H Paul D. Hanna, <a href="/A306404/b306404.txt">Table of n, a(n) for n = 0..300</a>

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

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

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

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

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

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

%F 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

%e 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! + ...

%e RELATED SERIES.

%e 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! + ...

%e 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! + ...

%o (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)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A322738.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 08 2019