A224381 Table of coefficients in the expansion of product((1+d_i*x), d_i|n).

1, 1, 1, 1, 3, 2, 1, 4, 3, 1, 7, 14, 8, 1, 6, 5, 1, 12, 47, 72, 36, 1, 8, 7, 1, 15, 70, 120, 64, 1, 13, 39, 27, 1, 18, 97, 180, 100, 1, 12, 11, 1, 28, 287, 1400, 3444, 4032, 1728, 1, 14, 13, 1, 24, 163, 336, 196, 1, 24, 158, 360, 225, 1, 31, 310, 1240, 1984, 1024

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..1500, flattened

Formula

T(n,k) = [x^k] Product_{d|n} (1+d*x).

Example

Row n = 6 : 1, 12, 47, 72, 36 because  (1+x)*(1+2x)*(1+3x)*(1+6x) = 1 + 12*x + 47*x^2 + 72*x^3 + 36*x^4.
Table begins :
  1;
  1,  1;
  1,  3,   2;
  1,  4,   3;
  1,  7,  14,    8;
  1,  6,   5;
  1, 12,  47,   72,   36;
  1,  8,   7;
  1, 15,  70,  120,   64;
  1, 13,  39,   27;
  1, 18,  97,  180,  100;
  1, 12,  11;
  1, 28, 287, 1400, 3444, 4032, 1728;
  1, 14,  13;
  1, 24, 163,  336,  196;
  1, 24, 158,  360,  225;
  1, 31, 310, 1240, 1984, 1024;
  ...

Maple

with(numtheory):
T:= proc(n) local p;
      p:= mul(1+d*x, d=divisors(n));
      seq(coeff(p, x, k), k=0..degree(p))
    end:
seq(T(n), n=0..30);  # Alois P. Heinz, Apr 05 2013

Mathematica

T[n_] := CoefficientList[Product[1+d*x, {d, Divisors[n]}], x]; T[0] = {1};
Array[T, 20, 0] // Flatten (* Jean-François Alcover, Mar 27 2017 *)

Crossrefs

Columns k=0-3 give: A000012, A000203, A119616, A067817.

Row lengths are: A000005(n)+1.

Last elements of rows give: A007955.

Sequence in context: A350499 A006020 A294177 * A190704 A346611 A190698

Adjacent sequences: A224378 A224379 A224380 * A224382 A224383 A224384

Keyword

nonn,look,tabf

Author

Philippe Deléham, Apr 05 2013

Status

approved