A143547 G.f. satisfies: A(x) = 1 + x*A(x)^4*A(-x)^3.

1, 1, 1, 4, 7, 34, 70, 368, 819, 4495, 10472, 59052, 141778, 814506, 1997688, 11633440, 28989675, 170574723, 430321633, 2552698720, 6503352856, 38832808586, 99726673130, 598724403680, 1547847846090, 9335085772194, 24269405074740, 146936230074004, 383846168712104

Offset

0,4

Comments

Number of achiral noncrossing partitions composed of n blocks of size 7. - 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.: A(x) = G(x^2) + x*G(x^2)^4 where G(x^2) = A(x)*A(-x) and G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

a(2n) = binomial(7*n,n)/(6*n+1); a(2n+1) = binomial(7*n+3,n)*4/(6*n+4).

G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2.

Example

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 34*x^5 + 70*x^6 + 368*x^7 + ...
Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then
A(x)*A(-x) = G(x^2) and A(x) = G(x^2) + x*G(x^2)^4 where
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 + ...
G(x)^4 = 1 + 4*x + 34*x^2 + 368*x^3 + 4495*x^4 + 59052*x^5 + ...
form the bisections of A(x).
By definition, A(x) = 1 + x*A(x)^4*A(-x)^3 where
A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 95*x^4 + 332*x^5 + 1074*x^6 + ...
A(-x)^3 = 1 - 3*x + 6*x^2 - 19*x^3 + 51*x^4 - 183*x^5 + 550*x^6 -+ ...

Mathematica

terms = 26;
A[_] = 1; Do[A[x_] = 1 + x A[x]^4 A[-x]^3 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

Prog

(PARI) {a(n)=my(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^4*subst(A^3, x, -x)); polcoef(A, n)}
(PARI) {a(n)=my(m=n\2, p=3*(n%2)+1); binomial(7*m+p-1, m)*p/(6*m+p)}

Crossrefs

Column k=7 of A369929 and k=8 of A370062.

Cf. A002296 (bisection), A143546.

Sequence in context: A243863 A153062 A237424 * A149090 A103059 A123809

Adjacent sequences: A143544 A143545 A143546 * A143548 A143549 A143550

Keyword

nonn

Author

Paul D. Hanna, Aug 23 2008

Extensions

a(26) onwards from Andrew Howroyd, Feb 08 2024

Status

approved