cp's OEIS Frontend

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

%I A212072 #12 May 13 2022 18:39:10
%S A212072 1,3,21,190,1950,21576,250971,3025308,37456650,473498025,6085977381,
%T A212072 79296104784,1044955576496,13903071489300,186507160795350,
%U A212072 2519857658331576,34258270557555282,468322722628414290,6433538749783033350,88767899653496377050,1229626632793564911906
%N A212072 G.f. satisfies: A(x) = (1 + x*A(x)^2)^3.
%H A212072 Michael De Vlieger, <a href="/A212072/b212072.txt">Table of n, a(n) for n = 0..855</a>
%H A212072 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H A212072 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%F A212072 a(n) = 3*binomial(6*n+3,n)/(6*n+3).
%F A212072 G.f. A(x) = G(x)^3 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
%e A212072 G.f.: A(x) = 1 + 3*x + 21*x^2 + 190*x^3 + 1950*x^4 + 21576*x^5 + ...
%e A212072 Related expansions:
%e A212072 A(x)^2 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 + ... + A002295(n+1)*x^n + ...
%e A212072 A(x)^(1/3) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + ... + A002295(n)*x^n + ...
%t A212072 Table[(3 Binomial[#, n])/# &[6 n + 3], {n, 0, 20}] (* _Michael De Vlieger_, May 13 2022 *)
%o A212072 (PARI) {a(n)=binomial(6*n+3,n) * 3/(6*n+3)}
%o A212072 for(n=0, 40, print1(a(n), ", "))
%o A212072 (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)}
%Y A212072 Cf. A002295, A212071, A212073, A130564.
%K A212072 nonn
%O A212072 0,2
%A A212072 _Paul D. Hanna_, Apr 29 2012