A183070 G.f.: A(x) = exp( Sum_{n>=1,k>=0} CATALAN(n,k)^2*x^(n+k)/n ), where CATALAN(n,k) = n*C(n+2*k-1,k)/(n+k) is the coefficient of x^k in C(x)^n and C(x) is the g.f. of the Catalan numbers.

1, 1, 2, 8, 49, 380, 3400, 33469, 352763, 3914105, 45203847, 539095203, 6600723606, 82616454685, 1053503618516, 13650703465841, 179351890161617, 2385294488375623, 32066177447127597, 435218601202213040

Offset

0,3

Comments

Compare the g.f. of this sequence to the g.f. of the Catalan numbers:

C(x) = exp( Sum_{n>=1,k>=0} C(n+k-1,k)^2*x^(n+k)/n ).

Links

Table of n, a(n) for n=0..19.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 49*x^4 + 380*x^5 + 3400*x^6 +...
The logarithm of the g.f. (A183069) begins:
log(A(x)) = x + 3*x^2/2 + 19*x^3/3 + 163*x^4/4 + 1626*x^5/5 +...
and equals the series:
log(A(x)) = (1 + x + 2^2*x^2 + 5^2*x^3 + 14^2*x^4 +...)*x
+ (1 + 2^2*x + 5^2*x^2 + 14^2*x^3 + 42^2*x^4 +...)*x^2/2
+ (1 + 3^2*x + 9^2*x^2 + 28^2*x^3 + 90^2*x^4 +...)*x^3/3
+ (1 + 4^2*x + 14^2*x^2 + 48^2*x^3 + 165^2*x^4 +...)*x^4/4
+ (1 + 5^2*x + 20^2*x^2 + 75^2*x^3 + 275^2*x^4 +...)*x^5/5 +...
which consists of the squares of coefficients in powers of C(x),
where C(x) = 1 + x*C(x)^2 is g.f. of the Catalan numbers (A000108).
...
Compare the above series for log(A(x)) to log(C(x)):
log(C(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 +...)*x^2/2
+ (1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 +...)*x^3/3
+ (1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 35^2*x^4 +...)*x^4/4
+ (1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 +...)*x^5/5 +...
which consists of the squares of coefficients in powers of 1/(1-x).

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, (m*binomial(m+2*k-1, k)/(m+k))^2*x^k)*x^m/m)+x*O(x^n)), n)}

Crossrefs

Cf. A183069 (log), A000108, A009766.

Sequence in context: A355667 A366239 A364592 * A032116 A088181 A058864

Adjacent sequences: A183067 A183068 A183069 * A183071 A183072 A183073

Keyword

nonn

Author

Paul D. Hanna, Dec 23 2010

Status

approved