A355723 Row 3 of table A355721.

1, 2, 18, 218, 3194, 53890, 1019250, 21256090, 483426010, 11895873410, 314834663250, 8918883839450, 269367643864250, 8643467766472450, 293770652998691250, 10546424484691428250, 398914704362503668250, 15860639479547463637250, 661439858772303085871250, 28874834455755565593004250

Offset: 0

Author

Peter Bala, Jul 15 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. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355724 (row 4), A355725 (row 5).

Programs

Maple

n := 3: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f: A(x) = ( Sum_{k >= 0} d(k+3)/d(3)*x^k )/( Sum_{k >= 0} d(k+2)/d(2)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).

A(x) = 1/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - ... )))).

The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 5*x*A(x)^2 - (1 + 3*x)*A(x) + 1 = 0 with A(0) = 1.

Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - 11*x/(1 - ... - 2*n*x/(1 - (2*n+5)*x )))))))), a continued fraction of Stieltjes type.