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

%I A143554 #34 Jul 09 2025 10:15:04
%S A143554 1,1,1,5,9,55,117,775,1785,12350,29799,211876,527085,3818430,9706503,
%T A143554 71282640,184138713,1366368375,3573805950,26735839650,70625252863,
%U A143554 531838637759,1416298046436,10723307329700,28748759731965,218658647805780,589546754316126
%N A143554 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x)^4.
%C A143554 Number of achiral noncrossing partitions composed of n blocks of size 9. - _Andrew Howroyd_, Feb 08 2024
%H A143554 Andrew Howroyd, <a href="/A143554/b143554.txt">Table of n, a(n) for n = 0..500</a>
%H A143554 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 A143554 G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.
%F A143554 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.
%F A143554 a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).
%F A143554 a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+2*j+2*k+2*l+2*m=n-1} a(i) * a(2*j) * a(2*k) * a(2*l) * a(2*m). - _Seiichi Manyama_, Jul 07 2025
%F A143554 a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (-1)^(x_1+x_2+x_3+x_4) * Product_{k=1..9} a(x_k). - _Seiichi Manyama_, Jul 09 2025
%e A143554 G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
%e A143554 Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then
%e A143554 G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
%e A143554 G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
%e A143554 G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...
%t A143554 terms = 25;
%t A143554 A[_] = 1; Do[A[x_] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
%t A143554 CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 24 2018 *)
%o A143554 (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)}
%o A143554 (PARI) {a(n)=my(m=n\2,p=4*(n%2)+1);binomial(9*m+p-1,m)*p/(8*m+p)}
%Y A143554 Column k=9 of A369929 and k=10 of A370062.
%Y A143554 Cf. A143338, A143546, A143547, A143550, A062994 (bisection).
%Y A143554 Cf. A143551, A143552, A143553.
%Y A143554 Cf. A143047.
%K A143554 nonn
%O A143554 0,4
%A A143554 _Paul D. Hanna_, Aug 24 2008