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A098348
Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.
4
1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 16, 48, 132, 308, 541, 32, 112, 368, 1076, 2612, 4683, 64, 256, 976, 3408, 10404, 25988, 47293, 128, 576, 2496, 10096, 36848, 116180, 296564, 545835, 256, 1280, 6208, 28480, 120400, 454608, 1469892, 3816548
OFFSET
1,2
COMMENTS
The first three columns are A000079, A001792 and A098385.
The first two diagonals are A000670 and A005649.
A070175 gives the smallest representative of each hook-type prime signature, so this sequence is a rearrangement of A074206(A070175).
FORMULA
a(n, k) = 1 + (Sum_{i=1..k-1} binomial(k-1, i)*a(i, i)) + (Sum_{j=1..k} Sum_{i=j..j+n-k-1} binomial(k-1, j-1)*a(i, j)) + (Sum_{j=1..k-1} binomial(k-1,j-1)*a(j+n-k, j)). - David Wasserman, Feb 21 2008
a(n, k) = A074206(2^(n+1-k)*A070826(k)). - David Wasserman, Feb 21 2008
The following conjectural formula for the triangle entries agrees with the values listed above: T(n,k) = Sum_{j = 0..n-k} 2^(n-k-j)*binomial(n-k,j)*a(k,j), where a(k,j) = 2^j*Sum_{i = j+1..k+1} binomial(i,j+1)*(i-1)!*Stirling2(k+1,i). See A098384 for related conjectures. - Peter Bala, Apr 20 2012
EXAMPLE
a(4, 2) = 20 because 24=2*2*2*3 has 20 ordered factorizations and so does any other number with the same prime signature.
CROSSREFS
Cf. A050324, A070175, A070826, A074206, A095705. A098349 gives the row sums. A098384.
Sequence in context: A356188 A238962 A238975 * A131420 A095705 A034776
KEYWORD
nonn,tabl,easy
AUTHOR
Alford Arnold, Sep 04 2004
EXTENSIONS
Edited and extended by David Wasserman, Feb 21 2008
STATUS
approved