A232692 - OEIS

%I #19 Nov 03 2019 04:55:12

%S 1,1,3,24,213,3096,46071,967608,20251809,555747048,15004870731,

%T 508165972056,16810393586733,677183788645704,26523956467895103,

%U 1238567261126084856,56056407696184372281,2976966230117448265128,152872356339113679491859,9098430770913969095416728

%N E.g.f. satisfies: A(x) = exp( 1/A(x)^3 * Integral A(x)^8 dx ).

%C Compare e.g.f. to: B(x) = exp( 1/B(x)^3 * Integral B(x)^3 dx ) where B(y) = Bessel polynomial y_n(-3) (cf. A065923).

%C Note that G(x) = exp(1/G(x)^3 * Integral G(x)^7 dx) has negative coefficients.

%C CONJECTURE:

%C Given G(x,n,k) = G such that G = exp( 1/G^n * Integral G^k dx ) then G(x,n,k) consists solely of positive coefficients when k >= A047399(n) where A047399 lists numbers that are congruent to {0,3,6} mod 8.

%H Paul D. Hanna, <a href="/A232692/b232692.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: (3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5). - _Vaclav Kotesovec_, Jan 05 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 213*x^4/4! + 3096*x^5/5! +...

%e Related expansions:

%e log(A(x)) = x + 2*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1905*x^5/5! + 23640*x^6/6! +...

%e Integral A(x)^8 dx = x + 8*x^2/2! + 80*x^3/3! + 1032*x^4/4! + 16320*x^5/5! +...

%e 1/A(x)^3 = 1 - 3*x + 3*x^2/2! - 24*x^3/3! + 117*x^4/4! - 2088*x^5/5! +...

%p seq(n! * coeff(series((3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5), x, n+1), x, n), n=0..20) # _Vaclav Kotesovec_, Jan 05 2014

%t m = 20; A[_] = 1; Do[A[x_] = Exp[1/A[x]^3 Integrate[A[x]^8 + O[x]^m, x]] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] Range[0, m-1]! (* _Jean-François Alcover_, Nov 03 2019 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(1/A^3*intformal(A^8+x*O(x^n))));n!*polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A232690, A232691.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 06 2013