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A143547
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G.f. satisfies: A(x) = 1 + x*A(x)^4*A(-x)^3.
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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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REFERENCES
| Bousquet, Michel; and Lamathe, Cedric; 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
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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) = C(7*n,n)/(6*n+1); a(2n+1) = C(7*n+3,n)*4/(6*n+4).
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2.
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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 -+...
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PROG
| (PARI) {a(n)=local(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^4*subst(A^3, x, -x)); polcoeff(A, n)}
(PARI) {a(n)=local(m=n\2, p=3*(n%2)+1); binomial(7*m+p-1, m)*p/(6*m+p)}
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CROSSREFS
| Cf. A002296 (bisection).
Sequence in context: A149089 A004031 A153062 * A149090 A103059 A123809
Adjacent sequences: A143544 A143545 A143546 * A143548 A143549 A143550
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2008
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