OFFSET
0,2
COMMENTS
Row sums of A101508.
Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009
LINKS
M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms
FORMULA
a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020
EXAMPLE
From Gus Wiseman, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
[4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
[1] [3] [2] [1 1]
[3] [1] [2] [1 1]
.
[1] [1] [2]
[1] [2] [1]
[2] [1] [1]
.
[1]
[1]
[1]
[1]
(End)
MAPLE
bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40); # Alois P. Heinz, Jan 30 2011
MATHEMATICA
a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)
PROG
(PARI) A101509(n) = sum( k=0, n, numdiv(k+1)*binomial(n, k)) \\ M. F. Hasler, Jan 14 2009
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 05 2004
STATUS
approved