A355796 Row 3 of A355793.

1, 3, 42, 786, 17736, 459768, 13333488, 425600976, 14791250688, 555381292800, 22398626084352, 965768866650624, 44347055502428160, 2161455366606034944, 111489317304231616512, 6069676735484389779456, 347921629212782938472448, 20950823605616500202323968, 1322561808699778749456678912

Offset

0,2

Links

Table of n, a(n) for n=0..18.

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

Formula

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

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 ).

A(x)/(1 - 8*x*A(x)) = Sum_{k >= 0} t(k+3)/t(3)*x^k.

A(x) = 1/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - 17*x/(1 + 17*x - ... )))) (continued fraction).

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.

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.

Maple

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);

Crossrefs

Cf. A355793 (table).

Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355797 (row 4).

Cf. A008544, A111528, A355721.

Sequence in context: A097068 A269046 A092470 * A336572 A366006 A366010

Adjacent sequences: A355793 A355794 A355795 * A355797 A355798 A355799

Keyword

nonn,easy

Author

Peter Bala, Jul 21 2022

Status

approved