A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) } for n>=1.

1, 2, 9, 68, 710, 9348, 148085, 2740672, 58033953, 1383923040, 36705564368, 1071911496576, 34179790156473, 1181725089179936, 44035415728886145, 1759481180119564288, 75042973200676887772, 3402984761691650083008

Offset

1,2

Comments

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

For n>=1, n divides a(n): a(n)/n = A120022(n).

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^(i-1) * C(n-j+i, j-i) * C(n-j+i-1, i-1) ];

a(n) = Sum_{j=0..n-1} n^j * [ Sum_{i=j..n-1} (-1)^(i-j) * Catalan(n-i-1) * C(n-i+j, i-j) * C(n-i+j-1, j) ], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

Example

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

Prog

(PARI) {a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j, (-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-1)))}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A120010, A120019, A120021, A120022.

Sequence in context: A364337 A186183 A295239 * A200248 A180747 A227457

Adjacent sequences: A120017 A120018 A120019 * A120021 A120022 A120023

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2006

Status

approved