A212072 G.f. satisfies: A(x) = (1 + x*A(x)^2)^3.

1, 3, 21, 190, 1950, 21576, 250971, 3025308, 37456650, 473498025, 6085977381, 79296104784, 1044955576496, 13903071489300, 186507160795350, 2519857658331576, 34258270557555282, 468322722628414290, 6433538749783033350, 88767899653496377050, 1229626632793564911906

Offset: 0

Author

Paul D. Hanna, Apr 29 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 190*x^3 + 1950*x^4 + 21576*x^5 + ...
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 + ... + A002295(n+1)*x^n + ...
A(x)^(1/3) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + ... + A002295(n)*x^n + ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..855
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Crossrefs

Cf. A002295, A212071, A212073, A130564.

Programs

Mathematica

Table[(3 Binomial[#, n])/# &[6 n + 3], {n, 0, 20}] (* Michael De Vlieger, May 13 2022 *)

PARI

{a(n)=binomial(6*n+3,n) * 3/(6*n+3)}
for(n=0, 40, print1(a(n), ", "))

PARI

{a(n)=local(A=1+3*x); for(i=1, n, A=(1+x*A^2)^3+x*O(x^n)); polcoeff(A, n)}

Formula

a(n) = 3*binomial(6*n+3,n)/(6*n+3).

G.f. A(x) = G(x)^3 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.