OFFSET
0,2
COMMENTS
See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.
LINKS
Robert Israel, Table of n, a(n) for n = 0..730
FORMULA
G.f.: A(x) = 1 + Series_Reversion((1+9*x - (1+x)^5)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (8+8*x)^(4*n+1)/9^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-1 + 9*sqrt(3)/(10*5^(1/4)))^(1/2 - n) / (3^(3/4) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
D-finite with recurrence: 80*(5*n + 11)*(5*n + 7)*(5*n + 3)*(5*n - 1)*a(n) + 10000*(2*n + 3)*(n + 1)*(10*n^2 + 30*n + 17)*a(n + 1) + 30000*(n + 2)*(n + 1)*(10*n^2 + 40*n + 39)*a(n + 2) + 100000*(2*n + 5)*(n + 3)*(n + 2)*(n + 1)*a(n + 3) - 9049*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) = 0. - Robert Israel, Mar 24 2026
EXAMPLE
A(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 1770*x^4 + 29208*x^5 +...
A(x)^5 = 1 + 10*x + 90*x^2 + 1080*x^3 + 15930*x^4 + 262872*x^5 +...
MAPLE
f:= gfun:-rectoproc({80*(5*n + 11)*(5*n + 7)*(5*n + 3)*(5*n - 1)*a(n) + 10000*(2*n + 3)*(n + 1)*(10*n^2 + 30*n + 17)*a(n + 1) + 30000*(n + 2)*(n + 1)*(10*n^2 + 40*n + 39)*a(n + 2) + 100000*(2*n + 5)*(n + 3)*(n + 2)*(n + 1)*a(n + 3) - 9049*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 10, a(3) = 120}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 24 2026
MATHEMATICA
CoefficientList[1 + InverseSeries[Series[(1+9*x - (1+x)^5)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
PROG
(PARI) {a(n)=local(A=1+2*x+10*x^2+x*O(x^n)); for(i=0, n, A=A+(-9*A+8+8*x+A^5)/4); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
EXTENSIONS
More terms from Robert Israel, Mar 24 2026
STATUS
approved