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%I #4 Jul 18 2022 19:47:58
%S 1,2,22,314,5326,102722,2197558,51355514,1297759918,35208930050,
%T 1020115715542,31432396066106,1026506419425550,35428218801977666,
%U 1288967076156307702,49323199246104202874,1980947315202528449518,83342865788161594337282,3666525676611059535630742
%N Row 4 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+4)/d(4)*x^k )/( Sum_{k >= 0} d(k+3)/d(3)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
%F A(x) = 1/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - 13*x/(1 + 13*x - ... )))).
%F The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 7*x*A(x)^2 - (1 + 5*x)*A(x) + 1 = 0 with A(0) = 1.
%F Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - 13*x/(1 - ... - 2*n*x/(1 - (2*n+7)*x )))))))), a continued fraction of Stieltjes-type.
%p n := 4: 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), A355723 (row 3), A355725 (row 5).
%K nonn,easy
%O 0,2
%A _Peter Bala_, Jul 15 2022
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