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1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, 76246683534, 1968284351178, 54576250392726, 1618348891438242, 51122453577462414, 1714406473587300138, 60843580566100937046, 2278637898592632599682, 89818339421620249242894, 3717488491001699691500298
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..19.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
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FORMULA
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O.g.f: A(x) = ( Sum_{k >= 0} d(k+2)/d(2)*x^k )/( Sum_{k >= 0} d(k+1)/d(1)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x)= 1/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 3*x*A(x)^2 - (1 + x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - 9*x/(1 - ... - 2*n*x/(1 - (2*n+3)*x )))))))), a continued fraction of Stieltjes type.
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MAPLE
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n := 2: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);
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CROSSREFS
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Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355723 (row 3), A355724 (row 4), A355725 (row 5).
Sequence in context: A317356 A336182 A224729 * A303395 A301271 A245267
Adjacent sequences: A355719 A355720 A355721 * A355723 A355724 A355725
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KEYWORD
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nonn,easy
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AUTHOR
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Peter Bala, Jul 15 2022
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STATUS
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approved
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