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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 Alois P. Heinz, Table of n, a(n) for n = 0..200 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 Adjacent sequences:  A327677 A327678 A327679 * A327681 A327682 A327683 KEYWORD nonn AUTHOR Alois P. Heinz, Sep 21 2019 STATUS approved

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Last modified June 18 11:58 EDT 2021. Contains 345098 sequences. (Running on oeis4.)