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A143601
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Number of labeled odd degree trees with 2n+1 nodes.
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4
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1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| E.g.f. satisfies: A(x) = cosh(x*A(x)).
E.g.f.: A(x) = (1/x)*Series_Reversion( x/cosh(x) ).
E.g.f.: sqrt(A(x)^2 - 1) = e.g.f. of A007106.
E.g.f.: exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.
E.g.f.: A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
E.g.f.: A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Aug 29 2008: (Start)
E.g.f. satisfies: A(x/cosh(x)) = cosh(x).
a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)
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EXAMPLE
| E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
The e.g.f. of A007106 (a bisection of A058014) is given by:
sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
The e.g.f. of A058014 is given by:
F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).
The e.g.f. of A143600 is given by:
G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.
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PROG
| (PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=cosh(x*A)); n!*polcoeff(A, n)}
(PARI) {a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1), 2*n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 29 2008]
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CROSSREFS
| Cf. A058014, A143600, A007106.
Sequence in context: A174564 A030256 A023332 * A203360 A050286 A096761
Adjacent sequences: A143598 A143599 A143600 * A143602 A143603 A143604
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 26 2008, May 27 2009
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EXTENSIONS
| Edited by Paul D. Hanna (pauldhanna(AT)juno.com), May 27 2009
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