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A256384
Number A(n,k) of factorizations of m^n into at most n factors, where m is a product of exactly k distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
4
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 19, 5, 1, 1, 1, 41, 171, 74, 7, 1, 1, 1, 122, 1675, 1975, 248, 11, 1, 1, 1, 365, 16683, 64182, 20096, 814, 15, 1, 1, 1, 1094, 166699, 2203215, 2213016, 187921, 2457, 22, 1, 1, 1, 3281, 1666731, 76727374, 268446852, 69406700, 1609727, 7168, 30, 1
OFFSET
0,9
COMMENTS
A(n,k) is also the number of k-partite partitions of (n)^k into at most n k-tuples. A(2,2) = 5: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(1,2),(1,0)], [(1,1),(1,1)].
EXAMPLE
A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into at most 2 factors: 36, 2*18, 3*12, 4*9, 6*6.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 2, 5, 14, 41, 122, ...
1, 3, 19, 171, 1675, 16683, ...
1, 5, 74, 1975, 64182, 2203215, ...
1, 7, 248, 20096, 2213016, 268446852, ...
MATHEMATICA
b[n_, k_, i_] := b[n, k, i] = If[n>k, 0, 1] + If[PrimeQ[n] || i<2, 0, Sum[ If[d > k, 0, b[n/d, d, i-1]], {d, Divisors[n][[2 ;; -2]]}]]; A[0, _] = 1; A[1, _] = 1; A[_, 0] = 1; A[n_, k_] := With[{t = Times @@ Prime[ Range[k] ]}, b[t^n, t^n, n]]; Table[diag = Table[A[n-k, k], {k, n, 0, -1}]; Print[ diag]; diag, {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
CROSSREFS
Columns k=0-3 give: A000012, A000041, A254686, A254811.
Rows n=0+1,2-3 give: A000012, A007051, A256493.
Cf. A219727.
Sequence in context: A008550 A064094 A090182 * A111673 A121391 A375555
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 27 2015
STATUS
approved