A355797 - OEIS

%I #8 Aug 15 2022 10:28:13

%S 1,3,51,1119,29103,859143,28091463,1002057591,38606468343,

%T 1595167432599,70315835952471,3293268346004439,163337193581191575,

%U 8554718468806548951,471976737725208306327,27369722655919760451159,1664858070989667129693975,106029602841882346657155543

%N Row 4 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 Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).

%F O.g.f.: A(x) = ( Sum_{k >= 0} t(k+4)/t(4)*x^k )/( Sum_{k >= 0} t(k+3)/t(3)*x^k ).

%F A(x)/(1 - 11*x*A(x)) = Sum_{k >= 0} t(k+4)/t(4)*x^k.

%F A(x) = 1/(1 + 11*x - 14*x/(1 + 14*x -17*x/(1 + 17*x - 20*x/(1 + 20*x - ... )))) (continued fraction).

%F A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 11*x*A(x)^2 - (1 + 8*x)*A(x) + 1 = 0 with A(0) = 1.

%F Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 14*x/(1 - 6*x/(1 - 17*x/(1 - 9*x/(1 - 20*x/(1 - 12*x/(1 - 23*x/(1 - ...))))))))), a continued fraction of Stieltjes type.

%p n := 4: 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), A355794 (row 1), A355795 (row 2), A355796 (row 3).

%Y Cf. A008544, A111528, A355721.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Jul 21 2022