A355794 Row 1 of A355793.

1, 3, 24, 282, 4236, 76548, 1608864, 38488152, 1032125136, 30670171248, 1000637672064, 35571839009952, 1368990872569536, 56720594992438848, 2517761078627172864, 119222916630934484352, 5999613754698100628736, 319763269764299852744448, 17994913747767982690289664

Offset: 0

Author

Peter Bala, Jul 19 2022

Keywords

• nonn
• easy

Links

• A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Crossrefs

Cf. A355793 (table).

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

Cf. A008544, A111528, A355721.

Programs

Maple

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);

Formula

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).

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

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

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.

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.