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

%I A143547 #38 Jul 08 2025 07:41:30
%S A143547 1,1,1,4,7,34,70,368,819,4495,10472,59052,141778,814506,1997688,
%T A143547 11633440,28989675,170574723,430321633,2552698720,6503352856,
%U A143547 38832808586,99726673130,598724403680,1547847846090,9335085772194,24269405074740,146936230074004,383846168712104
%N A143547 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x)^3.
%C A143547 Number of achiral noncrossing partitions composed of n blocks of size 7. - _Andrew Howroyd_, Feb 08 2024
%H A143547 Andrew Howroyd, <a href="/A143547/b143547.txt">Table of n, a(n) for n = 0..500</a>
%H A143547 Michel Bousquet and Cédric Lamathe, <a href="https://doi.org/10.46298/dmtcs.420">On symmetric structures of order two</a>, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176, See Table 1. - From _N. J. A. Sloane_, Jul 12 2011
%F A143547 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.
%F A143547 a(2n) = binomial(7*n,n)/(6*n+1); a(2n+1) = binomial(7*n+3,n)*4/(6*n+4).
%F A143547 G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2.
%F A143547 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
%F A143547 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
%e A143547 G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 34*x^5 + 70*x^6 + 368*x^7 + ...
%e A143547 Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then
%e A143547 A(x)*A(-x) = G(x^2) and A(x) = G(x^2) + x*G(x^2)^4 where
%e A143547 G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 + ...
%e A143547 G(x)^4 = 1 + 4*x + 34*x^2 + 368*x^3 + 4495*x^4 + 59052*x^5 + ...
%e A143547 form the bisections of A(x).
%e A143547 By definition, A(x) = 1 + x*A(x)^4*A(-x)^3 where
%e A143547 A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 95*x^4 + 332*x^5 + 1074*x^6 + ...
%e A143547 A(-x)^3 = 1 - 3*x + 6*x^2 - 19*x^3 + 51*x^4 - 183*x^5 + 550*x^6 -+ ...
%t A143547 terms = 26;
%t A143547 A[_] = 1; Do[A[x_] = 1 + x A[x]^4 A[-x]^3 + O[x]^terms // Normal, {terms}];
%t A143547 CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 24 2018 *)
%o A143547 (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)}
%o A143547 (PARI) {a(n)=my(m=n\2,p=3*(n%2)+1);binomial(7*m+p-1,m)*p/(6*m+p)}
%Y A143547 Column k=7 of A369929 and k=8 of A370062.
%Y A143547 Cf. A002296 (bisection), A143546.
%Y A143547 Cf. A143549, A143550.
%Y A143547 Cf. A143046, A143553.
%K A143547 nonn
%O A143547 0,4
%A A143547 _Paul D. Hanna_, Aug 23 2008
%E A143547 a(26) onwards from _Andrew Howroyd_, Feb 08 2024