login
A101509
Binomial transform of tau(n) (see A000005).
21
1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773
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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 05 2004
STATUS
approved