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

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, 6492, 1, 1, 14, 126, 910, 5250, 22722, 65562, 1, 1, 16, 168, 1456, 10500, 60592, 262248, 756688, 1, 1, 18, 216, 2184, 18900, 136332, 786744, 3405096, 9825030, 1

Offset: 0

Author

Alonso del Arte, Dec 22 2011

Keywords

• nonn
• tabl
• easy

Comments

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.

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.

Examples

			Triangle starts:
  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, 6492, 1;
		

Links

• Jinyuan Wang, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000629 (row sums).

Programs

Maple

A202687 := proc(n,k)
    if k = 0 or k = n then
        1;
    else
        binomial(n,k)*add(procname(k,j),j=0..k) ;
    end if;
end proc: # R. J. Mathar, Mar 15 2013

Mathematica

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]

Formula

T(n, k) = binomial(n, k)*Sum_{j=0..k} T(k, j).