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Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001222(n)), giving the number of divisors of n with m prime factors (counted with multiplicity).
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%I #22 Feb 25 2015 11:20:15

%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,

%T 2,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,2,2,1,1,2,1,1,2,1,1,1,1,2,2,2,

%U 1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,1,3,3,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,2

%N Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001222(n)), giving the number of divisors of n with m prime factors (counted with multiplicity).

%C All rows are palindromic. T(n,0) = T(n,A001222(n)) = 1.

%C Two numbers have identical rows in the table if and only if they have the same prime signature.

%C If n is a perfect square then Sum_{even m} T(n,m) = 1 + Sum_{odd m} T(n,m), otherwise Sum_{even m} T(n,m) = Sum_{odd m} T(n,m). - _Geoffrey Critzer_, Feb 08 2015

%H Alois P. Heinz, <a href="/A146291/b146291.txt">Rows n = 1..2500, flattened</a>

%H Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial calculator</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Roundness.html">Roundness</a>

%H G. Xiao, WIMS server, <a href="http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html">Factoris</a> (both expands and factors polynomials)

%F If the canonical factorization of n into prime powers is the product of p^e(p), then T(n,m) is the coefficient of k^m in the polynomial expansion of Product_p (sum_{i=0..e} k^i).

%e Rows begin:

%e 1;

%e 1, 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 1;

%e 1, 1, 1, 1;

%e 1, 1, 1;

%e 1, 2, 1;

%e ...

%e 12 has 1 divisor with 0 total prime factors (1), 2 with 1 (2 and 3), 2 with 2 (4 and 6) and 1 with 3 (12), for a total of 6. The 12th row of the table therefore reads (1, 2, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 2k^2 + (1)k^3 = (1 + k + k^2)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).

%p with(numtheory):

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(

%p add(x^bigomega(d), d=divisors(n))):

%p seq(T(n), n=1..100); # _Alois P. Heinz_, Feb 25 2015

%t Join[{{1}},

%t Table[nn = DivisorSigma[0, n];

%t CoefficientList[

%t Series[Product[(1 - x^i)/(1 - x), {i,

%t FactorInteger[n][[All, 2]] + 1}], {x, 0, nn}], x], {n, 2, 100}]] (* _Geoffrey Critzer_, Jan 01 2015 *)

%Y Row sums equal A000005(n). T(n,1) = A001221(n) for n>1.

%Y Row n of A007318 is identical to row A002110(n) of this table and also identical to the row for any squarefree number with n prime factors.

%Y Cf. A146292, A146289, A146290.

%K nonn,tabf

%O 1,12

%A _Matthew Vandermast_, Nov 11 2008