A192948 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^5 - 1)^n.
1, 1, 5, 60, 885, 14605, 258126, 4778340, 91460415, 1795377600, 35946770255, 731245323256, 15070729457030, 314011160190675, 6603561278126200, 139980599432879480, 2987856960226960551, 64162892863813071450, 1385270621375211268550
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5*x^2 + 60*x^3 + 885*x^4 + 14605*x^5 +... where (A(x) - 1)*(2 - A(x)^5) = x and A(x - 5*x^2 - 10*x^3 - 10*x^4 - 5*x^5 - x^6) = 1 + x. Related expansions. (A(x)^5-1) = 5*x + 35*x^2 + 410*x^3 + 6030*x^4 + 99376*x^5 +... (A(x)^5-1)^2 = 25*x^2 + 350*x^3 + 5325*x^4 + 89000*x^5 +... (A(x)^5-1)^3 = 125*x^3 + 2625*x^4 + 49125*x^5 + 925625*x^6 +... (A(x)^5-1)^4 = 625*x^4 + 17500*x^5 + 388750*x^6 + 8177500*x^7 +...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..725
- Vaclav Kotesovec, Recurrence
Programs
-
Mathematica
CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^5,{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 17 2013 *) -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^5-1+x*O(x^n))^m));polcoeff(A,n)} -
PARI
{a(n)=local(A=1+serreverse(2*x-x*(1+x)^5+x^2*O(x^n)));polcoeff(A,n)}
Formula
G.f. A(x) equals the formal inverse of function (x-1)*(2-x^5).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^5).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^5 ).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 23.55179049164234... is the root of the equation -46656 - 385675*d - 1217000*d^2 - 1775000*d^3 - 1100000*d^4 + 50000*d^5 = 0 and c = 0.03694932533778987590415707531... - Vaclav Kotesovec, Sep 17 2013
Comments