A120021 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-(n+1)*x) o (x-x^2) } for n>=1.

1, 3, 16, 130, 1416, 19236, 312512, 5906502, 127313320, 3082645951, 82848394752, 2447576485341, 78846484722208, 2750891289611235, 103344880800464896, 4159577854374314795, 178587276548655542112, 8147334149686335230068

Offset

1,2

Comments

Secondary diagonal of A120019, the table of self-compositions of A120010.

Links

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

Formula

a(n) = Sum_{j=1..n} Catalan(n-j)*[Sum_{i=1..j} (-1)^(j-i)*(n+1)^(i-1)*C(n-j+i, j-i)*C(n-j+i-1, i-1)], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

Example

Successive self-compositions of F(x), the g.f. of A120010, begin:
F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = (1)x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + (3)x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 68x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (130)x^4 + 710x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + (1416)x^5 + 9348x^6+..

Prog

(PARI) a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n, x)
(PARI) /* Alternative Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)*sum(i=1, j, (-1)^(j-i)*(n+1)^(i-1)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)))

Crossrefs

Cf. A120010, A120019, A120020.

Sequence in context: A368791 A135752 A218827 * A223897 A131490 A121673

Adjacent sequences: A120018 A120019 A120020 * A120022 A120023 A120024

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2006

Status

approved