A186195 Expansion of (1+5x+sqrt(1+2x+9x^2))/(2(1+2x)).

1, 1, 0, -2, 2, 6, -18, -6, 114, -146, -490, 1794, 266, -12986, 20958, 56778, -255774, 39390, 1853478, -3687918, -7441158, 42252726, -20345490, -293463462, 708206802, 1002083406, -7527677898, 6140678434, 48978210794, -142206136026, -127715768578

Offset

0,4

Comments

Hankel transform is A186196. Hankel transform of a(n+1) is (-2)^C(n+1,2).

Links

Table of n, a(n) for n=0..30.

Formula

D-finite with recurrence n*a(n) +(4*n-3)*a(n-1) +(13*n-33)*a(n-2) +18*(n-3)*a(n-3)=0. - R. J. Mathar, Feb 13 2015

From Peter Bala, Nov 08 2022: (Start)

O.g.f. A(x) = 1 + series reversion of x*(1 + x)/((1 - x)*(1 + 2*x)).

The g.f. satisfies the differential equation (1 + 4*x + 13*x^2 + 18*x^3)*A'(x) + (1 - 7*x)*A(x) + (2*x - 2) = 0 with A(0) = 1. Mathar's recurrence above follows from this. (End)

Mathematica

CoefficientList[Series[(1+5x+Sqrt[1+2x+9x^2])/(2(1+2x)), {x, 0, 30}], x] (* Harvey P. Dale, Dec 17 2021 *)

Crossrefs

Cf. A114710, A186196.

Sequence in context: A216971 A019575 A178882 * A256215 A253284 A176754

Adjacent sequences: A186192 A186193 A186194 * A186196 A186197 A186198

Keyword

sign,easy

Author

Paul Barry, Feb 14 2011

Status

approved