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%I #37 Apr 21 2020 17:31:24
%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,5,1,1,1,1,3,10,17,1,1,1,1,3,23,
%T 93,73,1,1,1,1,4,40,465,1417,388,1,1,1,1,4,73,1746,19834,32152,2461,1,
%U 1,1,1,5,114,5741,190131,1532489,1016489,18155,1,1
%N Number A(n,k) of factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Also number of ways to partition the multiset consisting of k copies each of 1, 2, ..., n into n multisets of size k.
%H Andrew Howroyd, <a href="/A257463/b257463.txt">Antidiagonals n = 0..27, flattened</a> (antidiagonals 0..12 from Alois P. Heinz)
%H P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-17/1/25.extract">Combinations derived from m identical sets of n different letters and their connexion with general magic squares</a>, Proc. London Math. Soc., 17 (1917), 25-41. The array is on page 40.
%H Math StackExchange, <a href="http://math.stackexchange.com/questions/1641433/">Number of ways to partition 40 balls with 4 colors into 4 baskets</a>
%H Marko Riedel, <a href="/A257463/a257463.txt">Maple program to compute array from cycle indices</a>
%e A(4,2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4.
%e A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.
%e A(2,4) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 2, 3, 3, 4, ...
%e 1, 1, 5, 10, 23, 40, 73, ...
%e 1, 1, 17, 93, 465, 1746, 5741, ...
%e 1, 1, 73, 1417, 19834, 190131, 1398547, ...
%e 1, 1, 388, 32152, 1532489, 43816115, 848597563, ...
%p with(numtheory):
%p b:= proc(n, i, k) option remember; `if`(n=1, 1,
%p add(`if`(d>i or bigomega(d)<>k, 0,
%p b(n/d, d, k)), d=divisors(n)))
%p end:
%p A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k$2, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..8);
%t b[n_, i_, k_] := b[n, i, k] = If[n==1, 1, DivisorSum[n, If[#>i || PrimeOmega[#] != k, 0, b[n/#, #, k]]&]];
%t A[n_, k_] := b[p = Product[Prime[i], {i, 1, n}]^k, p, k];
%t Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Mar 20 2017, translated from Maple *)
%Y Columns k=0+1, 2-4 give: A000012, A002135, A254243, A268668.
%Y Rows n=0+1, 2-5 give: A000012, A008619, A257464, A253259, A253263.
%Y Main diagonal gives A334286.
%Y Cf. A257462, A257493 (ordered factorizations).
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Apr 24 2015
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