A120596 G.f. satisfies: 6*A(x) = 5 + x + A(x)^5, starting with [1,1,10].
1, 1, 10, 210, 5505, 161601, 5082420, 167451780, 5705082795, 199354509755, 7105393162010, 257312347583330, 9440808323869455, 350189693739455535, 13110655796699158800, 494772468434359266960, 18801468275832345890970
Offset: 0
Keywords
Examples
A(x) = 1 + x + 10*x^2 + 210*x^3 + 5505*x^4 + 161601*x^5 +... A(x)^5 = 1 + 5*x + 60*x^2 + 1260*x^3 + 33030*x^4 + 969606*x^5 +...
Programs
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Mathematica
CoefficientList[1 + InverseSeries[Series[1+6*x - (1+x)^5, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *) -
PARI
{a(n)=local(A=1+x+10*x^2+x*O(x^n));for(i=0,n,A=A-6*A+5+x+A^5);polcoeff(A,n)}
Formula
G.f.: A(x) = 1 + Series_Reversion(1+6*x - (1+x)^5). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (5+x)^(4*n+1)/6^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-5 + 4*(6/5)^(5/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
Comments