A120018 The third self-composition of A120010; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120010.

1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, 1187952, 6006171, 30710553, 158535975, 825143145, 4325320191, 22814398392, 120999555588, 644878190175, 3451975941243, 18550877091063, 100047282676491, 541314936448764

Offset

1,2

Comments

Row 3 of A120019, the square table of self-compositions of A120010.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Formula

G.f.: A(x) = (1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2) ))/2.

Recurrence: n*a(n) = 2*(5*n-6)*a(n-1) - (31*n-66)*a(n-2) + 42*(n-3)*a(n-3) - 21*(n-4)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(14*sqrt(21)-42)*((7+sqrt(21))/2)^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Example

A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 114*x^5 + 480*x^6 + 2157*x^7 +...
G(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
where G(x) is the g.f. of A120010 and G(G(G(x))) = A(x).

Mathematica

CoefficientList[Series[(1 - Sqrt[1 - 4 x (1-x) / (1 -3 x + 3 x^2)]) / x / 2,  {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Prog

(PARI) {a(n)=polcoeff((1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2+x*O(x^n)) ))/2, n)}

Crossrefs

Cf. A120010, A120017 (2nd self-composition), A120019.

Sequence in context: A350589 A308554 A055730 * A354120 A091353 A279199

Adjacent sequences: A120015 A120016 A120017 * A120019 A120020 A120021

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2006

Extensions

Typo in Mma program fixed by Vincenzo Librandi, May 22 2013

Status

approved