A258927 E.g.f. satisfies: A(x) = Integral 1 + A(x)^6 dx.
1, 720, 410572800, 4492717498368000, 348990783113936240640000, 118162808964225967251573964800000, 130226468530398571130647349959852032000000, 384446125794905598149974467971605129718661120000000, 2644398446216951886577241780697447635225293650237849600000000
Offset: 0
Keywords
Examples
E.g.f.: A(x) = x + 720*x^7/7! + 410572800*x^13/13! + 4492717498368000*x^19/19! +... where Series_Reversion(A(x)) = x - x^7/7 + x^13/13 - x^19/19 + x^25/25 +... Also, A(x) = S(x)/C(x) where S(x) = x - 120*x^7/7! - 21859200*x^13/13! - 131273353728000*x^19/19! +...+ A258926(n)*x^(6*n+1)/(6*n+1)! +... C(x) = 1 - 120*x^6/6! - 21859200*x^12/12! - 131273353728000*x^18/18! +...+ A258926(n)*x^(6*n)/(6*n)! +... such that C(x)^6 + S(x)^6 = 1.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..75
Crossrefs
Programs
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PARI
/* E.g.f. Series_Reversion( Integral 1/(1+x^6) dx ): */ {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)} for(n=0, 20, print1(a(n), ", ")) -
PARI
/* E.g.f. A(x) = Integral 1 + A(x)^6 dx.: */ {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)} for(n=0, 20, print1(a(n), ", "))
Formula
E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^6) dx ).
(2) A(x)^3 = tan( 3 * Integral A(x)^2 dx ).
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:
(3) A(x) = S(x)/C(x),
(4) A(x) = Integral 1/C(x)^6 dx,
(5) A(x)^3 = S(x)^3/C(x)^3 = tan( 3 * Integral S(x)^2/C(x)^2 dx ).
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
Comments