A369528 Expansion of g.f. (1 - sqrt(1 - 2*x + 2*x*sqrt(1 - 4*x)))/(2*x).

0, 1, 1, 3, 7, 21, 62, 197, 637, 2123, 7196, 24807, 86608, 305792, 1089810, 3915789, 14169457, 51592561, 188888760, 694946257, 2568035734, 9527251374, 35472258506, 132502639491, 496423643176, 1864942753690, 7023753418030, 26514295486956, 100305236628550, 380218561128958

Offset

0,4

Links

Table of n, a(n) for n=0..29.

Carles Cardó, Growth and density in free magmas, arXiv:2401.07827 [math.CO], 2024. See p. 16.

Formula

G.f.: (1 - sqrt(1 - 2*x + 2*x*sqrt(1 - 4*x)))/(2*x).

a(n) ~ 2^(2*n - 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 28 2024

D-finite with recurrence n*(n+1)*a(n) -2*n*(7*n-8)*a(n-1) +4*(17*n^2-55*n+39)*a(n-2) +4*(-28*n^2+158*n-213)*a(n-3) +12*(-8*n^2+36*n-25)*a(n-4) +8*(56*n^2-460*n+921)*a(n-5) -16*(4*n-21)*(4*n-23)*a(n-6)=0. - R. J. Mathar, Mar 25 2024

Mathematica

CoefficientList[Series[(1 - Sqrt[1 - 2*x + 2*x*Sqrt[1 - 4*x]])/(2*x), {x, 0, 12}], x]

Crossrefs

Cf. A000108.

Sequence in context: A129366 A270049 A166358 * A148670 A148671 A148672

Adjacent sequences: A369525 A369526 A369527 * A369529 A369530 A369531

Keyword

nonn

Author

Michael De Vlieger, Jan 25 2024

Status

approved