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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.
3
1, 1, 5, 45, 1065, 61753, 9705069, 4394516773, 5931440509137, 24154079629381105, 300121111037478706517, 11510717148660156841731485, 1369013994385630011763634779641, 505666129597215709912984823873504809, 582167751341290615329122568805084839847101
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 12 2019
STATUS
approved