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A369528
Expansion of g.f. (1 - sqrt(1 - 2*x + 2*x*sqrt(1 - 4*x)))/(2*x).
0
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
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
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jan 25 2024
STATUS
approved