A130524 Diagonal immediately below the main diagonal of square array A130523.

1, 4, 18, 87, 442, 2332, 12677, 70605, 401172, 2317683, 13578615, 80502845, 482140580, 2912954129, 17733375207, 108676158775, 669914021414, 4151053001800, 25841001981211, 161534820531068, 1013566626969398, 6381398103680604, 40301852983776764, 255249505209864803, 1620819715569894894

Offset: 0

Author

Paul D. Hanna, Jun 02 2007

Keywords

• nonn

Crossrefs

Cf. A130523; diagonals: A007857, A130525; related: A000108, A001764.

Programs

PARI

{a(n) = my(C,F,D); C=Ser(vector(n+1,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(n+1,r,binomial(3*r-3,r-1)/(2*r-1))); D=1/(1-x*C*F-x*F^2); polcoef(C*D*F+x*O(x^n),n,x)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = C(x)*D(x)*F(x), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).

Extensions

Edited and corrected by Paul D. Hanna, Jan 27 2025