A258927 - OEIS

%I #21 Jun 18 2015 16:35:34

%S 1,720,410572800,4492717498368000,348990783113936240640000,

%T 118162808964225967251573964800000,

%U 130226468530398571130647349959852032000000,384446125794905598149974467971605129718661120000000,2644398446216951886577241780697447635225293650237849600000000

%N E.g.f. satisfies: A(x) = Integral 1 + A(x)^6 dx.

%C From _Vaclav Kotesovec_, Jun 17 2015: (Start)

%C In general, for k>2, if e.g.f. satisfies A(x) = Integral 1 + A(x)^k dx, then a(n) ~ k^(k/(k-1)) * n^(1/(k-1)) * (k*n)! * (k*sin(Pi/k)/Pi)^(k*n + k/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1))).

%C (End)

%H Vaclav Kotesovec, <a href="/A258927/b258927.txt">Table of n, a(n) for n = 0..75</a>

%F E.g.f. A(x) satisfies:

%F (1) A(x) = Series_Reversion( Integral 1/(1+x^6) dx ).

%F (2) A(x)^3 = tan( 3 * Integral A(x)^2 dx ).

%F Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^6)^(1/6) dx ) is the e.g.f. of A258926, then e.g.f. A(x) of this sequence satisfies:

%F (3) A(x) = S(x)/C(x),

%F (4) A(x) = Integral 1/C(x)^6 dx,

%F (5) A(x)^3 = S(x)^3/C(x)^3 = tan( 3 * Integral S(x)^2/C(x)^2 dx ).

%F a(n) ~ 2^(6/5) * 3^(6*n+12/5) * (6*n)! * n^(1/5) / (5^(1/5) * Gamma(1/5) * Pi^(6*n+6/5)). - _Vaclav Kotesovec_, Jun 18 2015

%e E.g.f.: A(x) = x + 720*x^7/7! + 410572800*x^13/13! + 4492717498368000*x^19/19! +...

%e where Series_Reversion(A(x)) = x - x^7/7 + x^13/13 - x^19/19 + x^25/25 +...

%e Also, A(x) = S(x)/C(x) where

%e S(x) = x - 120*x^7/7! - 21859200*x^13/13! - 131273353728000*x^19/19! +...+ A258926(n)*x^(6*n+1)/(6*n+1)! +...

%e C(x) = 1 - 120*x^6/6! - 21859200*x^12/12! - 131273353728000*x^18/18! +...+ A258926(n)*x^(6*n)/(6*n)! +...

%e such that C(x)^6 + S(x)^6 = 1.

%o (PARI) /* E.g.f. Series_Reversion( Integral 1/(1+x^6) dx ): */

%o {a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^6 + O(x^(6*n+2))) ) ); (6*n+1)!*polcoeff(A, 6*n+1)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* E.g.f. A(x) = Integral 1 + A(x)^6 dx.: */

%o {a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^6 + O(x^(6*n+2)) )); (6*n+1)!*polcoeff(A, 6*n+1)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A000182(n-1) (k=2), A258880 (k=3), A258901 (k=4), A258925 (k=5), A259112 (k=7), A259113 (k=8), A258926, A258994.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 15 2015