A308812 a(n) = Sum_{k=1..n} binomial(n,k) * floor(n/k).

1, 5, 13, 33, 61, 143, 246, 521, 985, 1995, 3499, 7923, 14028, 28642, 55603, 115369, 210665, 455399, 838338, 1755983, 3383652, 6974159, 13034492, 28011611, 52475486, 108821068, 210050941, 436273458, 824191369, 1744975533, 3301974301, 6867107913, 13250454241

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..3318

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k=1..n} binomial(n,k) * x^k/(1 - x^k).

a(n) = Sum_{k=1..n} Sum_{d|k} binomial(n,d).

a(n) ~ 3 * 2^(n-1). - Vaclav Kotesovec, May 28 2021

Maple

f:= proc(n) local k; add(binomial(n, k)*floor(n/k), k=1..n) end proc:
map(f, [$1..100]); # Robert Israel, Aug 23 2019

Mathematica

Table[Sum[Binomial[n, k] Floor[n/k] , {k, 1, n}], {n, 1, 33}]
Table[SeriesCoefficient[1/(1 - x) Sum[Binomial[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 33}]
Table[Sum[Sum[Binomial[n, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 33}]

Crossrefs

Cf. A056045.

Sequence in context: A272161 A272828 A370754 * A321124 A001981 A141025

Adjacent sequences: A308809 A308810 A308811 * A308813 A308814 A308815

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 22 2019

Status

approved