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%I #39 Dec 08 2020 08:37:29
%S 1,1,4,15,44,135,456,1239,3424,8694,27240,65846,171864,406133,960848,
%T 2615460,5998416,14304089,32273100,72271516,153768520,385905072,
%U 817485768,1841794483,3915726528,8388036950,17125197336,35051814558,78986793592,160176485813
%N Number of partitions of n into colored blocks of equal parts with colors from a set of size n.
%H Alois P. Heinz, <a href="/A321880/b321880.txt">Table of n, a(n) for n = 0..1650</a>
%F a(n) = [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j).
%F a(n) = A321884(n,n).
%F a(n) = Sum_{i=0..floor((sqrt(1+8*n)-1)/2)} n!/(n-i)! * A321878(n,i).
%F a(n) = n * A325916(n) for n > 0, a(n) = 1.
%e a(3) = 15: 3a, 3b, 3c, 2a1a, 2a1b, 2a1c, 2b1a, 2b1b, 2b1c, 2c1a, 2c1b, 2c1c, 111a, 111b, 111c.
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, k*add(
%p (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i) +b(n, i-1, k)))
%p end:
%p a:= n-> b(n$3):
%p seq(a(n), n=0..31);
%t 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]]];
%t a[n_] := b[n, n, n];
%t a /@ Range[0, 31] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y Main diagonal of A321884.
%Y Cf. A000142, A003056, A321878, A325916.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Aug 27 2019
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