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A120970
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G.f. satisfies: A(x/A(x)^2) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^2).
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5
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1, 1, 2, 9, 60, 504, 4946, 54430, 655362, 8496454, 117311198, 1711459903, 26228829200, 420370445830, 7021029571856, 121859518887327, 2192820745899978, 40831103986939664, 785429260324068156
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008: (Start)
More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)
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FORMULA
| G.f. satisfies: A(x) = 1 + x*B(x)^2 = 1 + (1 + x*C(x)^2 )^2 where B(x) and C(x) satisfy: C(x) = B(x)*B(A(x)-1), B(x) = A(A(x)-1), B(A(x)-1) = A(B(x)-1), B(x/A(x)^2) = A(x), B(x) = A(x*B(x)^2) and B(x) is g.f. of A120971.
a(n) = [x^(n-1)] (1/n)*A(x)^(2n) for n>=1 with a(0)=1; i.e., a(n) equals 1/n times the coefficient of x^(n-1) in A(x)^(2n) for n>=1. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008]
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PROG
| (PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=-Vec(subst(Ser(A), x, x/Ser(A)^2))[ #A]); A[n+1]}
(PARI) /* This sequence is generated when k=2, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n, k=2, m=0)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008]
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CROSSREFS
| Cf. A120971; variants: A120972, A120974, A120976, A030266, A067145, A107096.
Cf. related variants: A145347, A145348, A147664, A145349, A145350. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008]
Sequence in context: A156272 A205570 A116364 * A111558 A168449 A001193
Adjacent sequences: A120967 A120968 A120969 * A120971 A120972 A120973
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jul 20 2006
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