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A202687 Triangle arising in the computation of hypersigma, definition 2 (A191161).

%I A202687 #18 Jul 31 2021 14:28:27
%S A202687 1,1,1,1,4,1,1,6,18,1,1,8,36,104,1,1,10,60,260,750,1,1,12,90,520,2250,
%T A202687 6492,1,1,14,126,910,5250,22722,65562,1,1,16,168,1456,10500,60592,
%U A202687 262248,756688,1,1,18,216,2184,18900,136332,786744,3405096,9825030,1
%N A202687 Triangle arising in the computation of hypersigma, definition 2 (A191161).
%C A202687 Given a squarefree number with n distinct prime factors put through the hypersigma function (A191161), each smaller divisor contributes a given number of "loose" 1s. Row n of this triangle tells how many loose 1s are contributed by prime divisors (column 1), by semiprime divisors (column 2), sphenic divisors (column 3) and so on up to column n - 1.
%C A202687 It may appear somewhat artificial that the leftmost and rightmost columns are all filled with 1s since under the hypersigma function, the smallest divisor of a number, 1, may be said to contribute two loose 1s (itself and the 1 in the recursion level immediately below). However, the reason for attributing one of these 1s to the number itself and the 1 in the recursion level below to that 1 in the recursion level above is to more clearly show how this triangle is tied to Pascal's triangle.
%H A202687 Jinyuan Wang, <a href="/A202687/b202687.txt">Rows n = 0..100 of triangle, flattened</a>
%F A202687 T(n, k) = binomial(n, k)*Sum_{j=0..k} T(k, j).
%e A202687 Triangle starts:
%e A202687   1;
%e A202687   1,  1;
%e A202687   1,  4,  1;
%e A202687   1,  6, 18,   1;
%e A202687   1,  8, 36, 104,    1;
%e A202687   1, 10, 60, 260,  750,    1;
%e A202687   1, 12, 90, 520, 2250, 6492, 1;
%p A202687 A202687 := proc(n,k)
%p A202687     if k = 0 or k = n then
%p A202687         1;
%p A202687     else
%p A202687         binomial(n,k)*add(procname(k,j),j=0..k) ;
%p A202687     end if;
%p A202687 end proc: # _R. J. Mathar_, Mar 15 2013
%t A202687 a[0, k_] := 1; a[n_, n_] := 1; a[n_, k_] := a[n, k] = Binomial[n, k] Sum[a[k, j], {j, 0, k}]; ColumnForm[Table[a[n, k], {n, 0, 9}, {k, 0, n}], Center]
%Y A202687 Cf. A000629 (row sums).
%K A202687 nonn,tabl,easy
%O A202687 0,5
%A A202687 _Alonso del Arte_, Dec 22 2011