A254686
Number of ways to put n red and n blue balls into n indistinguishable boxes.
Original entry on oeis.org
1, 1, 5, 19, 74, 248, 814, 2457, 7168, 19928, 53688, 139820, 354987, 878434, 2128102, 5052010, 11781881, 27019758, 61035671, 135928105, 298784144, 648726349, 1392474574, 2956730910, 6214668074, 12937060340, 26686392239, 54572423946, 110680119454, 222710856175, 444776676764
Offset: 0
For n = 2 the a(2) = 5 ways to put 2 red balls and 2 blue balls into 2 indistinguishable boxes are (RRBB)(), (RRB)(B), (RBB)(R), (RR)(BB), (RB)(RB).
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with(numtheory):
b:= proc(n, k, i) option remember;
`if`(n>k, 0, 1) +`if`(isprime(n) or i<2, 0, add(
`if`(d>k, 0, b(n/d, d, i-1)), d=divisors(n) minus {1, n}))
end:
a:= n-> b(6^n$2,n):
seq(a(n), n=0..20); # Alois P. Heinz, Mar 26 2015
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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[n_] := b[6^n, 6^n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
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.
Original entry on oeis.org
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
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, ...
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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 *)
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