OFFSET
0,3
COMMENTS
Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..875
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
FORMULA
G.f. A(x) equals the formal inverse of function (x-1)*(2-x^3).
G.f. satisfies: A(x) = 1 + x/(2 - A(x)^3).
G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^3 ).
Recurrence: 108*(n-2)*(n-1)*n*(9*n-25)*a(n) = 18*(n-2)*(n-1)*(702*n^2 - 3003*n + 2924)*a(n-1) + 3*(n-2)*(3375*n^3 - 22875*n^2 + 50704*n - 36656)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(9*n-16)*a(n-3). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ 1/48*sqrt(6)*sqrt(1271 - (2661+32096*sqrt(2))^(2/3) + 3*(2661+32096*sqrt(2))^(1/3))/((2661 + 32096*sqrt(2))^(1/6)) * ((3*(841+34*sqrt(2))^(2/3) + 267 + 26*(841+34*sqrt(2))^(1/3)) / (841+34*sqrt(2))^(1/3)/6)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 181*x^4 + 1746*x^5 + 18039*x^6 +...
where (A(x) - 1)*(2 - A(x)^3) = x
and A(x - 3*x^2 - 3*x^3 - x^4) = 1+x.
Related expansions.
(A(x)^3-1) = 3*x + 12*x^2 + 82*x^3 + 705*x^4 + 6792*x^5 + 70122*x^6 +...
(A(x)^3-1)^2 = 9*x^2 + 72*x^3 + 636*x^4 + 6198*x^5 + 64396*x^6 +...
(A(x)^3-1)^3 = 27*x^3 + 324*x^4 + 3510*x^5 + 38475*x^6 +...
(A(x)^3-1)^4 = 81*x^4 + 1296*x^5 + 16632*x^6 + 203148*x^7 +...
Also,
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 705*x^4 + 6792*x^5 + 70122*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 124*x^3 + 1067*x^4 + 10284*x^5 + 106200*x^6 +...
where 2+x = 2*A(x) + A(x)^3 - A(x)^4.
MATHEMATICA
CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^3, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Sep 17 2013 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^3-1+x*O(x^n))^m)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+serreverse(2*x-x*(1+x)^3+x^2*O(x^n))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 13 2011
STATUS
approved