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..959
FORMULA
G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^4)/27).
G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x)^(3*n+1) / 13^(4*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(3*n - 7/3) * 3^(2*n) / (13^(1/3) * sqrt(Pi) * n^(3/2) * (2^(1/3)*13^(4/3) - 32)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017
D-finite with recurrence: -216*(4*n + 5)*(2*n + 1)*(4*n - 1)*a(n) - 192*(n + 1)*(48*n^2 + 96*n + 43)*a(n + 1) - 2048*(2*n + 3)*(n + 2)*(n + 1)*a(n + 2) + 451*(n + 1)*(n + 2)*(n + 3)*a(n + 3) = 0. - Robert Israel, Mar 25 2026
EXAMPLE
A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 249*x^4 + 1932*x^5 +...
A(x)^4 = 1 + 12*x + 78*x^2 + 468*x^3 + 3237*x^4 + 25116*x^5 +...
MAPLE
f:= gfun:-rectoproc({-216*(4*n + 5)*(2*n + 1)*(4*n - 1)*a(n) - 192*(n + 1)*(48*n^2 + 96*n + 43)*a(n + 1) - 2048*(2*n + 3)*(n + 2)*(n + 1)*a(n + 2) + 451*(n + 1)*(n + 2)*(n + 3)*a(n + 3), a(0) = 1, a(1) = 3, a(2) = 6}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 25 2026
MATHEMATICA
CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)
PROG
(PARI) {a(n)=local(A=1+3*x+6*x^2+x*O(x^n)); for(i=0, n, A=A+(-13*A+12+27*x+A^4)/9); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
EXTENSIONS
More terms from Robert Israel, Mar 25 2026
STATUS
approved