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 A219585 Number A(n,k) of k-partite partitions of {n}^k into distinct k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals. 14
 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 2, 1, 1, 15, 40, 17, 2, 1, 1, 52, 457, 364, 46, 3, 1, 1, 203, 6995, 14595, 2897, 123, 4, 1, 1, 877, 136771, 937776, 407287, 21369, 323, 5, 1, 1, 4140, 3299218, 88507276, 107652681, 10200931, 148257, 809, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS A(n,k) is the number of factorizations of m^n into distinct factors where m is a product of k distinct primes.  A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into distinct factors: 36, 3*12, 4*9, 2*18, 2*3*6. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..209 FORMULA A(n,k) = [(Product_{j=1..k} x_j)^n] 1/2 * Product_{i_1,...,i_k>=0} (1+Product_{j=1..k} x_j^i_j). EXAMPLE A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)]. A(3,2) = 17: [(3,3)], [(3,0),(0,3)], [(3,2),(0,1)], [(2,3),(1,0)], [(3,1),(0,2)], [(2,2),(1,1)], [(1,3),(2,0)], [(2,1),(1,2)], [(2,1),(1,1),(0,1)], [(3,0),(0,2),(0,1)], [(2,2),(1,0),(0,1)], [(2,1),(0,2),(1,0)], [(1,2),(2,0),(0,1)], [(1,2),(1,1),(1,0)], [(0,3),(2,0),(1,0)], [(2,0),(1,1),(0,2)], [(2,0),(0,2),(1,0),(0,1)]. Square array A(n,k) begins:   1,  1,   1,      1,          1,            1,         1, ...   1,  1,   2,      5,         15,           52,       203, ...   1,  1,   5,     40,        457,         6995,    136771, ...   1,  2,  17,    364,      14595,       937776,  88507276, ...   1,  2,  46,   2897,     407287,    107652681,  ...   1,  3, 123,  21369,   10200931,  10781201973,  ...   1,  4, 323, 148257,  233051939,  ...   1,  5, 809, 970246, 4909342744,  ... MATHEMATICA f[n_, k_] := f[n, k] = 1/2 Product[Sum[O[x[j]]^(n+1), {j, 1, k}]+1+ Product[x[j]^i[j], {j, 1, k}], Evaluate[Sequence @@ Table[{i[j], 0, n}, {j, 1, k}]]]; a[0, _] = a[_, 0] = 1; a[n_, k_] := SeriesCoefficient[f[n, k], Sequence @@ Table[{x[j], 0, n}, {j, 1, k}]]; Table[Print[a[n-k, k]]; a[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013, updated Sep 16 2019 *) PROG (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)} T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ Andrew Howroyd, Dec 16 2018 CROSSREFS Columns k=0..5 give: A000012, A000009, A219554, A219560, A219561, A219565. Rows n=0..3 give: A000012, A000110, A094574, A319591. Cf. A188445, A219727 (partitions of {n}^k into k-tuples), A318286. Sequence in context: A181783 A121395 A275377 * A292464 A090628 A054387 Adjacent sequences:  A219582 A219583 A219584 * A219586 A219587 A219588 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Nov 23 2012 STATUS approved

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Last modified May 9 03:39 EDT 2021. Contains 343685 sequences. (Running on oeis4.)