A143554 G.f. satisfies: A(x) = 1 + x*A(x)^5*A(-x)^4.

1, 1, 1, 5, 9, 55, 117, 775, 1785, 12350, 29799, 211876, 527085, 3818430, 9706503, 71282640, 184138713, 1366368375, 3573805950, 26735839650, 70625252863, 531838637759, 1416298046436, 10723307329700, 28748759731965, 218658647805780, 589546754316126

Offset

0,4

Comments

Number of achiral noncrossing partitions composed of n blocks of size 9. - Andrew Howroyd, Feb 08 2024

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1. - From N. J. A. Sloane, Jul 12 2011

Formula

G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.

G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).

Example

G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then
G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...

Mathematica

terms = 25;
A[_] = 1; Do[A[x_] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

Prog

(PARI) {a(n)=my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^5*subst(A^4, x, -x)); polcoef(A, n)}
(PARI) {a(n)=my(m=n\2, p=4*(n%2)+1); binomial(9*m+p-1, m)*p/(8*m+p)}

Crossrefs

Column k=9 of A369929 and k=10 of A370062.

Cf. A143338, A143546, A143547, A143550, A143551, A143552, A143553, A062994 (bisection).

Sequence in context: A262918 A283918 A284382 * A222536 A222698 A200440

Adjacent sequences: A143551 A143552 A143553 * A143555 A143556 A143557

Keyword

nonn

Author

Paul D. Hanna, Aug 24 2008

Status

approved