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A088717
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G.f. satisfies: A(x) = 1 + x*A(x)^2*A(x*A(x)^2).
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7
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1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971, 45640572, 526788153, 6392402793, 81247489335, 1078331283648, 14907041720241, 214187010762831, 3192620516380376, 49287883925072010, 786925082232918304
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Coefficient of x^n of A(x)^2 equals coefficient of x^n in (1+x*A(x))^(n+1)/(n+1).
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FORMULA
| Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xA^2*B, B = 1 + x(AB)^2*C, C = 1 + x(ABC)^2*D, D = 1 + x(ABCD)^2*E, etc., then B(x)=A(x*A(x)^2), C(x)=B(x*A(x)^2), D(x)=C(x*A(x)^2), etc., where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence (see table A128330).
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007
Recurrence:
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} m*C(2n+m,k)/(2n+m) * a(n-k,k). [Paul D. Hanna, Dec 16 2010]
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EXAMPLE
| Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + xAB;
B = A*(1 + xBC);
C = B*(1 + xCD);
D = C*(1 + xDE);
E = D*(1 + xEF) ; ...
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PROG
| (PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^2*subst(A, x, x*A^2+x*O(x^n))); polcoeff(A, n)}
(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(2*n+m, k)/(2*n+m)*a(n-k, k))))}
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CROSSREFS
| Cf. A128330, A030266.
Cf. A002449, A030266, A087949, A088714, A091713, A120971.
Sequence in context: A074535 A190761 A005700 * A111538 A160881 A088716
Adjacent sequences: A088714 A088715 A088716 * A088718 A088719 A088720
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KEYWORD
| nonn,eigen
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 12 2003 and Mar 10 2007
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