A051054 a(n) = Sum_{k=1..n} C(n, floor(n/k)).

0, 1, 3, 7, 15, 26, 54, 85, 159, 292, 513, 804, 1844, 2965, 5169, 10679, 20107, 34120, 72498, 126028, 245966, 498852, 913872, 1644570, 3600916, 6530881, 12280999, 25149973, 48355605, 89310576, 187976827, 348475899, 677303827

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) is asymptotic to 2^n/sqrt(n*Pi/2). - Benoit Cloitre, Jan 11 2003

Maple

A051054 := proc(n) local k; add(binomial(n, floor(n/k)), k=1..n); end; [seq(A051054(n), n=0..40)];

Mathematica

Table[Sum[Binomial[n, Floor[n/i]], {i, n}], {n, 0, 40}] (* Wesley Ivan Hurt, May 16 2016 *)

Prog

(PARI) a(n) = sum(k=1, n, binomial(n, n\k)); \\ Seiichi Manyama, Jan 06 2022
(Python)
from math import comb
def A051054(n):
    c, j = 0, 1
    while j <= n:
        c += comb(n, k:=n//j)*(-j+(j:=n//k+1))
    return c # Chai Wah Wu, Jan 28 2026

Crossrefs

Cf. A056045, A273160, A273161, A345466.

Sequence in context: A131076 A393145 A001648 * A001649 A303220 A301894

Adjacent sequences: A051051 A051052 A051053 * A051055 A051056 A051057

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved