A273160 -id:A273160 - cp's OEIS frontend

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

N. J. A. Sloane

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

Cf. A056045, A273160, A273161, A345466.

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

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

a(n) = sum(k=1, n, binomial(n, n\k)); \\ Seiichi Manyama, Jan 06 2022

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

A273161 a(n) = Sum_{k=1..n} C(n-k, floor((n-k)/k)).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 13, 19, 34, 61, 108, 174, 384, 641, 1166, 2337, 4458, 7828, 16421, 29346, 57231, 114126, 215915, 396491, 839324, 1549146, 2983185, 5978656, 11628952, 21812113, 45099914, 84842925, 166417181, 332267593, 647614074, 1234586894, 2538571022

Offset: 0

Wesley Ivan Hurt, May 16 2016

Cf. A051054, A273160.

A273161:=n->add(binomial(n-i,floor((n-i)/i)), i=1..n): seq(A273161(n), n=0..50);

Table[Sum[Binomial[n - i, Floor[(n - i)/i]], {i, n}], {n, 0, 40}]

a(n) ~ 2^(n - 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 28 2021

cp's OEIS Frontend

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

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