cp's OEIS Frontend

A355723 Row 3 of table A355721.

%I A355723 #5 Jul 18 2022 19:47:48
%S A355723 1,2,18,218,3194,53890,1019250,21256090,483426010,11895873410,
%T A355723 314834663250,8918883839450,269367643864250,8643467766472450,
%U A355723 293770652998691250,10546424484691428250,398914704362503668250,15860639479547463637250,661439858772303085871250,28874834455755565593004250
%N A355723 Row 3 of table A355721.
%H A355723 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 A355723 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 A355723 A(x) = 1/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - ... )))).
%F A355723 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 A355723 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 A355723 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 A355723 Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355724 (row 4), A355725 (row 5).
%K A355723 nonn,easy
%O A355723 0,2
%A A355723 _Peter Bala_, Jul 15 2022