A143547 G.f. A(x) 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

Author

Paul D. Hanna, Aug 23 2008

Keywords

• nonn

Comments

Number of achiral noncrossing partitions composed of n blocks of size 7. - Andrew Howroyd, Feb 08 2024

Examples

			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 -+ ...
		

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

Crossrefs

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

Cf. A002296 (bisection), A143546.

Cf. A143549, A143550.

Cf. A143046, A143553.

Programs

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 *)

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)}

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.

a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+2*j+2*k+2*l=n-1} a(i) * a(2*j) * a(2*k) * a(2*l). - Seiichi Manyama, Jul 07 2025

a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} (-1)^(x_1+x_2+x_3) * Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

Extensions

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