OFFSET
1,2
COMMENTS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 14 2006
STATUS
approved