A326650 Number of colored integer partitions using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

1, 1, 5, 45, 1065, 61753, 9705069, 4394516773, 5931440509137, 24154079629381105, 300121111037478706517, 11510717148660156841731485, 1369013994385630011763634779641, 505666129597215709912984823873504809, 582167751341290615329122568805084839847101

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

Maple

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= k-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), n=k..g(k)):
seq(a(n), n=0..15);

Mathematica

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k] Binomial[k, t]], {j, 0, n/i}]]];
a[k_] := Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {n, k, g[k]}, {i, 0, k}];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Crossrefs

Column sums of A326616 and of A326617.

Sequence in context: A211051 A322661 A191962 * A323572 A368491 A318092

Adjacent sequences: A326647 A326648 A326649 * A326651 A326652 A326653

Keyword

nonn

Author

Alois P. Heinz, Sep 12 2019

Status

approved