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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, 1, 2, 8, 49, 380, 3400, 33469, 352763, 3914105, 45203847, 539095203, 6600723606, 82616454685, 1053503618516, 13650703465841, 179351890161617, 2385294488375623, 32066177447127597, 435218601202213040 (list; graph; refs; listen; history; text; internal format)
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 L(x) to:

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: A009751 A279239 A279109 * A032116 A088181 A058864

Adjacent sequences:  A183067 A183068 A183069 * A183071 A183072 A183073

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 23 2010

STATUS

approved

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Last modified March 26 23:00 EDT 2019. Contains 321565 sequences. (Running on oeis4.)