A120014 Coefficients of x^n in the n-th iteration of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2 } for n>=1.

1, 2, 9, 60, 530, 5892, 79681, 1276760, 23729310, 502780580, 11974950746, 316917570312, 9230453871756, 293492484431720, 10117826259791025, 375952605020796720, 14980065429077943734, 637215061582781559972

Offset

1,2

Comments

a(n) is divisible by n for n>=1; a(n)/n = A120016(n).

Main diagonal of table (A120013) of iterations of A120009.

Links

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

Formula

a(n) = [x^n] x*((1-n+n^2) - n^2*(n+1)*x - n*(1-(n+2)*x)*C(x) )/(1-n+n^2*x)^2, where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

a(n) = n^(n-1) - Sum_{k=2..n-2} n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!/n!

Example

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

Prog

(PARI) a(n)=local(k=n, x=X+X^3*O(X^n)); polcoeff( x*((1-k+k^2)-k^2*(k+1)*x-k*(1-(k+2)*x)*(1-sqrt(1-4*x))/2/x)/(1-k+k^2*x)^2, n, X)
(PARI) /* Generated as the n-th self-composition of A120009: */ a(n)=local(F=((1-3*x)*sqrt(1-4*x+x^3*O(x^n)) - (1-x)*(1-4*x))/(2*x^2), G=x+x*O(x^n)); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n, x)))
(PARI) a(n)=n^(n-1)-sum(k=2, n-2, n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!)/n!

Crossrefs

Cf. A120016 (a(n)/n); A120009, A127275 (g.f.=F(F(x))), A120012 (g.f.=F(F(F(x)))); A000108 (Catalan); A120015, A120020, A120013.

Sequence in context: A168449 A001193 A161391 * A036774 A306065 A166882

Adjacent sequences: A120011 A120012 A120013 * A120015 A120016 A120017

Keyword

nonn

Author

Paul D. Hanna, Jun 07 2006, Jun 09 2006

Status

approved