A245269 Sum of binomial(n,k) over cubefree k.
1, 3, 7, 15, 31, 63, 127, 254, 502, 978, 1882, 3600, 6904, 13380, 26332, 52664, 106744, 218232, 447736, 917760, 1873312, 3799920, 7653136, 15306272, 30429856, 60234528, 118956831, 234885092, 464595690, 921868388, 1836393687, 3672648928, 7369572624, 14821243232
Offset: 1
Keywords
Links
- Eric M. Schmidt, Table of n, a(n) for n = 1..1000
- J. E. Nymann and W. J. Leahey, On the probability that an integer chosen according to the binomial distribution be k-free, Rocky Mountain Journal of Mathematics 7 (1977), no. 4, 769-774.
Programs
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Mathematica
cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; a[n_] := Sum[Binomial[n, k], {k, Select[Range[n], cubeFreeQ]}]; Array[a, 34] (* Amiram Eldar, May 25 2025 *) -
Sage
def A245269(n) : return sum(binomial(n,k) for k in range(1,n+1) if all(m <= 2 for (p,m) in factor(k)))
Formula
a(n) ~ 2^n/zeta(3). [Take p = 1/2 in Nymann and Leahey.]