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

%I A224381 #21 Mar 18 2023 17:31:41
%S A224381 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 A224381 1,13,39,27,1,18,97,180,100,1,12,11,1,28,287,1400,3444,4032,1728,1,14,
%U A224381 13,1,24,163,336,196,1,24,158,360,225,1,31,310,1240,1984,1024
%N A224381 Table of coefficients in the expansion of product((1+d_i*x), d_i|n).
%H A224381 Alois P. Heinz, <a href="/A224381/b224381.txt">Rows n = 0..1500, flattened</a>
%F A224381 T(n,k) = [x^k] Product_{d|n} (1+d*x).
%e A224381 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 A224381 Table begins :
%e A224381   1;
%e A224381   1,  1;
%e A224381   1,  3,   2;
%e A224381   1,  4,   3;
%e A224381   1,  7,  14,    8;
%e A224381   1,  6,   5;
%e A224381   1, 12,  47,   72,   36;
%e A224381   1,  8,   7;
%e A224381   1, 15,  70,  120,   64;
%e A224381   1, 13,  39,   27;
%e A224381   1, 18,  97,  180,  100;
%e A224381   1, 12,  11;
%e A224381   1, 28, 287, 1400, 3444, 4032, 1728;
%e A224381   1, 14,  13;
%e A224381   1, 24, 163,  336,  196;
%e A224381   1, 24, 158,  360,  225;
%e A224381   1, 31, 310, 1240, 1984, 1024;
%e A224381   ...
%p A224381 with(numtheory):
%p A224381 T:= proc(n) local p;
%p A224381       p:= mul(1+d*x, d=divisors(n));
%p A224381       seq(coeff(p, x, k), k=0..degree(p))
%p A224381     end:
%p A224381 seq(T(n), n=0..30);  # _Alois P. Heinz_, Apr 05 2013
%t A224381 T[n_] := CoefficientList[Product[1+d*x, {d, Divisors[n]}], x]; T[0] = {1};
%t A224381 Array[T, 20, 0] // Flatten (* _Jean-François Alcover_, Mar 27 2017 *)
%Y A224381 Columns k=0-3 give: A000012, A000203, A119616, A067817.
%Y A224381 Row lengths are: A000005(n)+1.
%Y A224381 Last elements of rows give: A007955.
%K A224381 nonn,look,tabf
%O A224381 0,5
%A A224381 _Philippe Deléham_, Apr 05 2013