A153295 G.f.: A(x) = F(x*G(x)^2) where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 4, 20, 110, 638, 3828, 23515, 146972, 930869, 5958094, 38462190, 250054804, 1635421543, 10750864640, 70987129653, 470542935654, 3129729034478, 20880459397920, 139689406647522, 936832986074664, 6297064070279195

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Formula

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.

G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^2 where G(x) is the g.f. of A001764.

G.f. satisfies: A(x/F(x)) = F(x*F(x)) where F(x) is the g.f. of A000108 (Catalan).

From Alexander Burstein, Nov 23 2019: (Start)

G.f. satisfies: A(x) = 1 + x*G(x)^3*m(x*G(x)^3), where m(x) is the g.f. of A001006 (Motzkin numbers) and G(x) is the g.f. of A001764 (ternary trees).

G.f. satisfies: A(-x*A(x)^7) = 1/A(x). (End)

Example

G.f.: A(x) = F(x*G(x)^2) = 1 + x + 4*x^2 + 20*x^3 + 110*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 276*x^4 + 1656*x^5 +...
G(x)^2*A(x)^2 = 1 + 4*x + 20*x^2 + 110*x^3 + 638*x^4 +...

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*binomial(3*(n-k)+2*k, n-k)*2*k/(3*(n-k)+2*k)))}

Crossrefs

Cf. A001764, A000108, A001006; A153294, A153296.

Sequence in context: A222205 A262394 A271932 * A006770 A158827 A263854

Adjacent sequences: A153292 A153293 A153294 * A153296 A153297 A153298

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved