A199137 G.f. satisfies: A(x) = exp( Sum_{n>=1} C(2*n,n)/2 * A(x^n) * x^n/n ).

1, 1, 3, 9, 30, 97, 336, 1153, 4081, 14552, 52609, 191657, 704385, 2604476, 9687433, 36207241, 135920489, 512182805, 1936656361, 7345211322, 27935373368, 106509551719, 407015199144, 1558603221623, 5979839952471, 22983021033071, 88477003979994, 341120527468590

Offset

0,3

Comments

Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x) = exp( Sum_{n>=1} C(2*n,n)/2 * x^n/n ).

Links

Table of n, a(n) for n=0..27.

Formula

G.f.: Product_{n>=1} C(x^n)^a(n-1) = Sum_{n>=0} a(n)*x^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 97*x^5 + 336*x^6 +...
where
log(A(x)) = A(x)*x + 3*A(x^2)*x^2/2 + 10*A(x^3)*x^3/3 + 35*A(x^4)*x^4/4 + 126*A(x^5)*x^5/5 + 462*A(x^6)*x^6/6 + 1716*A(x^7)*x^7/7 + 6435*A(x^8)*x^8/8 +...
The g.f. also equals the product:
A(x) = C(x) * C(x^2) * C(x^3)^3 * C(x^4)^9 * C(x^5)^30 * C(x^6)^97 *...* C(x^n)^a(n-1) *...
where C(x) is the g.f. of the Catalan numbers:
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A, x, x^m+x*O(x^n))*binomial(2*m, m)/2*x^m/m))); polcoeff(A, n)}

Crossrefs

Sequence in context: A337267 A337034 A250128 * A089978 A052906 A102898

Adjacent sequences: A199134 A199135 A199136 * A199138 A199139 A199140

Keyword

nonn

Author

Paul D. Hanna, Nov 03 2011

Status

approved