A224381 - OEIS

%I #21 Mar 18 2023 17:31:41

%S 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,

%T 1,13,39,27,1,18,97,180,100,1,12,11,1,28,287,1400,3444,4032,1728,1,14,

%U 13,1,24,163,336,196,1,24,158,360,225,1,31,310,1240,1984,1024

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

%H Alois P. Heinz, <a href="/A224381/b224381.txt">Rows n = 0..1500, flattened</a>

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

%e 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.

%e Table begins :

%e   1;

%e   1,  1;

%e   1,  3,   2;

%e   1,  4,   3;

%e   1,  7,  14,    8;

%e   1,  6,   5;

%e   1, 12,  47,   72,   36;

%e   1,  8,   7;

%e   1, 15,  70,  120,   64;

%e   1, 13,  39,   27;

%e   1, 18,  97,  180,  100;

%e   1, 12,  11;

%e   1, 28, 287, 1400, 3444, 4032, 1728;

%e   1, 14,  13;

%e   1, 24, 163,  336,  196;

%e   1, 24, 158,  360,  225;

%e   1, 31, 310, 1240, 1984, 1024;

%e   ...

%p with(numtheory):

%p T:= proc(n) local p;

%p       p:= mul(1+d*x, d=divisors(n));

%p       seq(coeff(p, x, k), k=0..degree(p))

%p     end:

%p seq(T(n), n=0..30);  # _Alois P. Heinz_, Apr 05 2013

%t T[n_] := CoefficientList[Product[1+d*x, {d, Divisors[n]}], x]; T[0] = {1};

%t Array[T, 20, 0] // Flatten (* _Jean-François Alcover_, Mar 27 2017 *)

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

%Y Row lengths are: A000005(n)+1.

%Y Last elements of rows give: A007955.

%K nonn,look,tabf

%O 0,5

%A _Philippe Deléham_, Apr 05 2013