cp's OEIS Frontend

A355796 Row 3 of A355793.

%I A355796 #8 Aug 15 2022 10:28:21
%S A355796 1,3,42,786,17736,459768,13333488,425600976,14791250688,555381292800,
%T A355796 22398626084352,965768866650624,44347055502428160,2161455366606034944,
%U A355796 111489317304231616512,6069676735484389779456,347921629212782938472448,20950823605616500202323968,1322561808699778749456678912
%N A355796 Row 3 of A355793.
%H A355796 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 A355796 Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
%F A355796 O.g.f.: A(x) = ( Sum_{k >= 0} t(k+3)/t(3)*x^k )/( Sum_{k >= 0} t(k+2)/t(2)*x^k ).
%F A355796 A(x)/(1 - 8*x*A(x)) = Sum_{k >= 0} t(k+3)/t(3)*x^k.
%F A355796 A(x) = 1/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - 17*x/(1 + 17*x - ... )))) (continued fraction).
%F A355796 A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 8*x*R(n,x)^2 - (1 + 5*x)*R(n,x) + 1 = 0 with A(0) = 1.
%F A355796 Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 11*x/(1 - 6*x/(1 - 14*x/(1 - 9*x/(1 - 17*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.
%p A355796 n := 3: 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 A355796 Cf. A355793 (table).
%Y A355796 Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355797 (row 4).
%Y A355796 Cf. A008544, A111528, A355721.
%K A355796 nonn,easy
%O A355796 0,2
%A A355796 _Peter Bala_, Jul 21 2022