A325916 Number of partitions of n into colored blocks of equal parts with colors from a set of size n such that the block with largest parts has the first color.

1, 1, 2, 5, 11, 27, 76, 177, 428, 966, 2724, 5986, 14322, 31241, 68632, 174364, 374901, 841417, 1792950, 3803764, 7688426, 18376432, 37158444, 80078021, 163155272, 335521478, 658661436, 1298215354, 2820956914, 5523327097, 11240000648, 22117134452, 43666070406

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1650

Formula

a(n) = 1/n * [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j) for n>0, a(0)=1.

a(n) = A321880(n)/n for n > 0, a(0) = 1.

Example

a(3) = 5: 3a, 2a1a, 2a1b, 2a1c, 111a.

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, k*add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i) +b(n, i-1, k)))
    end:
a:= n-> `if`(n=0, 1, b(n$3)/n):
seq(a(n), n=0..34);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, k Sum[With[{t = n - i j},  b[t, Min[t, i - 1], k]], {j, 1, n/i}] + b[n, i - 1, k]]];
a[n_] := If[n == 0, 1, b[n, n, n]/n];
a /@ Range[0, 34] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

Crossrefs

Cf. A321880.

Sequence in context: A027087 A363579 A055227 * A257790 A174145 A358451

Adjacent sequences: A325913 A325914 A325915 * A325917 A325918 A325919

Keyword

nonn

Author

Alois P. Heinz, Sep 08 2019

Status

approved