cp's OEIS Frontend

A355795 Row 2 of A355793.

%I A355795 #9 Aug 15 2022 10:28:17
%S A355795 1,3,33,507,9609,212835,5350785,149961675,4628365305,155913036915,
%T A355795 5692874399025,224034935130075,9456933847187625,426402330032719875,
%U A355795 20460268520575152225,1041301103429870128875,56040353252589013121625,3180443637298592493577875,189863589771186976073108625
%N A355795 Row 2 of A355793.
%H A355795 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 A355795 O.g.f.: A(x) = ( Sum_{k >= 0} t(k+2)/t(2)*x^k )/( Sum_{k >= 0} t(k+1)/t(1)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
%F A355795 A(x)/(1 - 5*x*A(x)) = Sum_{k >= 0} t(k+2)/t(2)*x^k.
%F A355795 A(x) = 1/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - ... )))) (continued fraction).
%F A355795 A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 5*x*A(x)^2 - (1 + 2*x)*A(x) + 1 = 0 with A(0) = 1.
%F A355795 Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 8*x/(1 - 6*x/(1 - 11*x/(1 - 9*x/(1 - 14*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.
%p A355795 n := 2: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);
%Y A355795 Cf. A355793 (table).
%Y A355795 Cf. A112936 (row 0), A355794 (row 1), A355796 (row 3), A355797 (row 4).
%Y A355795 Cf. A008544, A111528, A355721.
%K A355795 nonn,easy
%O A355795 0,2
%A A355795 _Peter Bala_, Jul 21 2022