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Row 3 of table A355721.
4

%I #5 Jul 18 2022 19:47:48

%S 1,2,18,218,3194,53890,1019250,21256090,483426010,11895873410,

%T 314834663250,8918883839450,269367643864250,8643467766472450,

%U 293770652998691250,10546424484691428250,398914704362503668250,15860639479547463637250,661439858772303085871250,28874834455755565593004250

%N Row 3 of table A355721.

%H A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

%F O.g.f: A(x) = ( Sum_{k >= 0} d(k+3)/d(3)*x^k )/( Sum_{k >= 0} d(k+2)/d(2)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).

%F A(x) = 1/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - ... )))).

%F The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 5*x*A(x)^2 - (1 + 3*x)*A(x) + 1 = 0 with A(0) = 1.

%F Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - 11*x/(1 - ... - 2*n*x/(1 - (2*n+5)*x )))))))), a continued fraction of Stieltjes type.

%p n := 3: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

%Y Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355724 (row 4), A355725 (row 5).

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 15 2022