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A327680
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Total number of colors used in all colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.
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2
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0, 1, 7, 44, 358, 2904, 29112, 296448, 3520568, 43482208, 602603120, 8712724080, 138736978208, 2302036052128, 41417364992160, 776413790063328, 15597709327298944, 325945020056535968, 7238587734613470208, 166897326948551436384, 4061690336695535982048
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} k * A309973(n,k).
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
end:
a:= n-> add(add(k*b(n$2, i)*(-1)^(k-i)*
binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..22);
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[k b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
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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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