A376556 The number of non-unitary square divisors of binomial(n, floor(n/2)).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 4, 6, 2, 2, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 6, 8, 0, 0, 0, 4, 4, 6, 2, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 4, 4, 0, 0, 4, 8, 2, 3, 6, 8

Offset: 0

Author

Amiram Eldar, Sep 28 2024

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001405, A056061, A056626, A376553, A376554, A376555.

Programs

Mathematica

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); a[0] = a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[Binomial[n, Floor[n/2]]]) - Times @@ f2 @@@ fct; Array[a, 60, 0]

PARI

a(n) = {my(e = factor(binomial(n, n\2))[, 2]); vecprod(apply(x -> x\2 + 1, e)) - vecprod(apply(x -> 1 << (1 - x%2), e));}

Formula

a(n) = A056626(A001405(n)).