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

Author

Alois P. Heinz, Sep 08 2019

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A321880.

Programs

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 *)

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.