A120015 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-(n+1)*x) o (1-sqrt(1-4*x))/2 } for n>=1.

1, 3, 16, 120, 1164, 13965, 200960, 3387636, 65644780, 1440018822, 35314018656, 958109355632, 28508766348664, 923461269689985, 32357613376995840, 1219728800410342556, 49225886778689380044, 2118029584754948604618

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Formula

a(n) = Sum_{j=1..n} (n+1)^(j-2)*(n-j+2)*j*(2*n-j-1)!/(n-j)!/n! - Paul D. Hanna and Max Alekseyev.

Example

Successive self-compositions of F(x), the g.f. of A120009, begin:
F(x) = x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
F(F(x)) = (1)x + 2x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
F(F(F(x))) = x + (3)x^2 + 9x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 60x^4 + 192x^5 + 360x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (120)x^4 + 530x^5 +1955x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 + (1164)x^5 +5892x^6+...

Prog

(PARI) {a(n)=sum(j=1, n, (n+1)^(j-2)*(n-j+2)*j*(2*n-j-1)!/(n-j)!/n!)}

Crossrefs

Cf. A120014; A120009, A127275 (g.f.=F(F(x))), A120012 (g.f.=F(F(F(x)))); A120020.

Sequence in context: A136168 A187735 A200318 * A003692 A166883 A145158

Adjacent sequences: A120012 A120013 A120014 * A120016 A120017 A120018

Keyword

nonn

Author

Paul D. Hanna, Jun 12 2006

Status

approved