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A325915
Total number of colors used in all colored integer partitions of n where all colors from an initial interval of the color palette are used and parts differ by size or by color.
3
0, 1, 3, 9, 25, 67, 176, 453, 1149, 2882, 7161, 17654, 43238, 105303, 255210, 615896, 1480771, 3548313, 8477415, 20199596, 48014369, 113879450, 269555798, 636875077, 1502195104, 3537705916, 8319377813, 19537936874, 45827441193, 107366261405, 251268532266
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=1..n} k * A308680(n,k).
a(n) ~ c * d^n * n, where d = 2.26562663992642295791262530033324290454663... is the root of the equation QPochhammer[-1, 1/d] = 4 and c = 0.1771510533646387556482103930322780317974659818141571819... - Vaclav Kotesovec, Sep 18 2019
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
g:= proc(n) option remember; `if`(n=0, [1, 0],
(p-> p+[0, p[1]])(add(b(j)*g(n-j), j=1..n)))
end:
a:= n-> g(n)[2]:
seq(a(n), n=0..32);
MATHEMATICA
b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
g[n_] := g[n] = If[n == 0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[b[j] g[n - j], {j, 1, n}]]];
a[n_] := g[n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 08 2019
STATUS
approved