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
Keywords
Examples
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).
Programs
-
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)}
Comments