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A101509 Binomial transform of tau(n) (see A000005). 17
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 (list; graph; refs; listen; history; text; internal format)
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

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

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

Sequence in context: A238913 A133124 A104004 * A240741 A240742 A240743

Adjacent sequences:  A101506 A101507 A101508 * A101510 A101511 A101512

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Dec 05 2004

STATUS

approved

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Last modified February 20 20:55 EST 2019. Contains 320345 sequences. (Running on oeis4.)