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.

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

Alois P. Heinz, Table of n, a(n) for n = 0..2807

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

Cf. A000009, A304969, A308680.

Sequence in context: A245748 A181383 A106514 * A268451 A156561 A085327

Adjacent sequences: A325912 A325913 A325914 * A325916 A325917 A325918

Keyword

nonn

Author

Alois P. Heinz, Sep 08 2019

Status

approved