A355794 - OEIS

%I #11 Aug 15 2022 10:28:26

%S 1,3,24,282,4236,76548,1608864,38488152,1032125136,30670171248,

%T 1000637672064,35571839009952,1368990872569536,56720594992438848,

%U 2517761078627172864,119222916630934484352,5999613754698100628736,319763269764299852744448,17994913747767982690289664

%N Row 1 of A355793.

%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} t(k+1)/t(1)*x^k )/( Sum_{k >= 0} t(k)/t(0)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).

%F A(x)/(1 - 2*x*A(x)) = Sum_{k >= 0} t(k+1)/t(1)*x^k.

%F A(x) = 1/(1 + 2*x - 5*x/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - ... )))) (continued fraction).

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

%F Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 5*x/(1 - 6*x/(1 - 8*x/(1 - 9*x/(1 - 11*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.

%p n := 1: 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 Cf. A355793 (table).

%Y Cf. A112936 (row 0), A355795 (row 2), A355796 (row 3), A355797 (row 4).

%Y Cf. A008544, A111528, A355721.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 19 2022