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..702
FORMULA
G.f.: A(x) = 1 + Series_Reversion((1+15*x - (1+x)^6)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (14+27*x)^(5*n+1)/15^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1/2 + 3*n) * (-14 + 5*(5/2)^(6/5))^(1/2 - n) / (2^(3/5) * 5^(9/10) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
D-finite with recurrence: -52488*(6*n - 1)*(3*n + 2)*(2*n + 3)*(3*n + 7)*(6*n + 19)*a(n) - 54432*(n + 1)*(1620*n^4 + 12960*n^3 + 36315*n^2 + 41580*n + 16024)*a(n + 1) - 1905120*(2*n + 5)*(n + 2)*(n + 1)*(24*n^2 + 120*n + 137)*a(n + 2) - 658560*(n + 3)*(n + 2)*(n + 1)*(72*n^2 + 432*n + 641)*a(n + 3) - 6146560*(2*n + 7)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) + 533607*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 5) = 0. - Robert Israel, Mar 23 2026
EXAMPLE
A(x) = 1 + 3*x + 15*x^2 + 210*x^3 + 3510*x^4 + 65562*x^5 +...
A(x)^6 = 1 + 18*x + 225*x^2 + 3150*x^3 + 52650*x^4 + 983430*x^5 +...
MAPLE
f:= gfun:-rectoproc({-52488*(6*n - 1)*(3*n + 2)*(2*n + 3)*(3*n + 7)*(6*n + 19)*a(n) - 54432*(n + 1)*(1620*n^4 + 12960*n^3 + 36315*n^2 + 41580*n + 16024)*a(n + 1) - 1905120*(2*n + 5)*(n + 2)*(n + 1)*(24*n^2 + 120*n + 137)*a(n + 2) - 658560*(n + 3)*(n + 2)*(n + 1)*(72*n^2 + 432*n + 641)*a(n + 3) - 6146560*(2*n + 7)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) + 533607*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 5), a(0) = 1, a(1) = 3, a(2) = 15, a(3) = 210, a(4) = 3510}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 23 2026
MATHEMATICA
CoefficientList[1 + InverseSeries[Series[(1+15*x - (1+x)^6)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
PROG
(PARI) {a(n)=local(A=1+3*x+15*x^2+x*O(x^n)); for(i=0, n, A=A+(-15*A+14+27*x+A^6)/9); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
EXTENSIONS
More terms from Robert Israel, Mar 23 2026
STATUS
approved