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A327680 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. 2
0, 1, 7, 44, 358, 2904, 29112, 296448, 3520568, 43482208, 602603120, 8712724080, 138736978208, 2302036052128, 41417364992160, 776413790063328, 15597709327298944, 325945020056535968, 7238587734613470208, 166897326948551436384, 4061690336695535982048 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=1..n} k * A309973(n,k).
MAPLE
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);
MATHEMATICA
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}];
a /@ Range[0, 22] (* Jean-François Alcover, Dec 18 2020, after_Alois P. Heinz_ *)
CROSSREFS
Cf. A309973.
Sequence in context: A091127 A166775 A221541 * A156374 A171493 A262772
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 21 2019
STATUS
approved

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)