A245268 Sum of binomial(n,k) over squarefree k.
1, 3, 7, 14, 26, 48, 92, 184, 375, 758, 1497, 2884, 5461, 10286, 19507, 37584, 73866, 147987, 301075, 618794, 1278116, 2640993, 5439593, 11138764, 22640100, 45644797, 91293390, 181301470, 358024924, 704359427, 1383415456, 2718141072, 5351701032, 10570658330
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
a[n_] := Sum[Binomial[n, k], {k, Select[Range[n], SquareFreeQ]}]; Array[a, 34] (* Amiram Eldar, May 25 2025 *) -
PARI
a(n) = sum(k=1, n, if (issquarefree(k), binomial(n,k), 0)); \\ Michel Marcus, Jul 16 2014
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Sage
def A235268(n) : return sum(binomial(n,k) for k in range(1,n+1) if is_squarefree(k))
Formula
a(n) ~ 2^n/zeta(2). [Take p = 1/2 in Nymann and Leahey.]