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A321880
Number of partitions of n into colored blocks of equal parts with colors from a set of size n.
3
1, 1, 4, 15, 44, 135, 456, 1239, 3424, 8694, 27240, 65846, 171864, 406133, 960848, 2615460, 5998416, 14304089, 32273100, 72271516, 153768520, 385905072, 817485768, 1841794483, 3915726528, 8388036950, 17125197336, 35051814558, 78986793592, 160176485813
OFFSET
0,3
LINKS
FORMULA
a(n) = [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j).
a(n) = A321884(n,n).
a(n) = Sum_{i=0..floor((sqrt(1+8*n)-1)/2)} n!/(n-i)! * A321878(n,i).
a(n) = n * A325916(n) for n > 0, a(n) = 1.
EXAMPLE
a(3) = 15: 3a, 3b, 3c, 2a1a, 2a1b, 2a1c, 2b1a, 2b1b, 2b1c, 2c1a, 2c1b, 2c1c, 111a, 111b, 111c.
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-> b(n$3):
seq(a(n), n=0..31);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
a[n_] := b[n, n, n];
a /@ Range[0, 31] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)
CROSSREFS
Main diagonal of A321884.
Sequence in context: A329523 A331317 A259664 * A075673 A062827 A074448
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 27 2019
STATUS
approved