A355797 Row 4 of A355793.

1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, 1595167432599, 70315835952471, 3293268346004439, 163337193581191575, 8554718468806548951, 471976737725208306327, 27369722655919760451159, 1664858070989667129693975, 106029602841882346657155543

Offset

0,2

Links

Table of n, a(n) for n=0..17.

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+4)/t(4)*x^k )/( Sum_{k >= 0} t(k+3)/t(3)*x^k ).

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

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

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.

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.

Maple

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);

Crossrefs

Cf. A355793 (table).

Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355796 (row 3).

Cf. A008544, A111528, A355721.

Sequence in context: A361051 A307369 A126685 * A246693 A187666 A377491

Adjacent sequences: A355794 A355795 A355796 * A355798 A355799 A355800

Keyword

nonn,easy

Author

Peter Bala, Jul 21 2022

Status

approved