OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 15 2009
STATUS
approved