

A202687


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


1



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, 1, 20, 270, 3120, 31500
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OFFSET

0,5


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.


LINKS

Table of n, a(n) for n=0..59.


FORMULA

T(n, k) = C(n, k) *sum_{j = 0 .. k} T(k, j)


EXAMPLE

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


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]


CROSSREFS

Cf. A000629, row sums.
Sequence in context: A069322 A208332 A075112 * A046554 A010321 A046550
Adjacent sequences: A202684 A202685 A202686 * A202688 A202689 A202690


KEYWORD

nonn,tabl,easy


AUTHOR

Alonso del Arte, Dec 22 2011


STATUS

approved



