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A326650
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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.
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3
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1, 1, 5, 45, 1065, 61753, 9705069, 4394516773, 5931440509137, 24154079629381105, 300121111037478706517, 11510717148660156841731485, 1369013994385630011763634779641, 505666129597215709912984823873504809, 582167751341290615329122568805084839847101
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OFFSET
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0,3
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LINKS
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MAPLE
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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);
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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