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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.

%I A325916 #20 Dec 15 2020 16:27:48
%S A325916 1,1,2,5,11,27,76,177,428,966,2724,5986,14322,31241,68632,174364,
%T A325916 374901,841417,1792950,3803764,7688426,18376432,37158444,80078021,
%U A325916 163155272,335521478,658661436,1298215354,2820956914,5523327097,11240000648,22117134452,43666070406
%N 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.
%H A325916 Alois P. Heinz, <a href="/A325916/b325916.txt">Table of n, a(n) for n = 0..1650</a>
%F A325916 a(n) = 1/n * [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j) for n>0, a(0)=1.
%F A325916 a(n) = A321880(n)/n for n > 0, a(0) = 1.
%e A325916 a(3) = 5: 3a, 2a1a, 2a1b, 2a1c, 111a.
%p A325916 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, k*add(
%p A325916       (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i) +b(n, i-1, k)))
%p A325916     end:
%p A325916 a:= n-> `if`(n=0, 1, b(n$3)/n):
%p A325916 seq(a(n), n=0..34);
%t A325916 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]]];
%t A325916 a[n_] := If[n == 0, 1, b[n, n, n]/n];
%t A325916 a /@ Range[0, 34] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y A325916 Cf. A321880.
%K A325916 nonn
%O A325916 0,3
%A A325916 _Alois P. Heinz_, Sep 08 2019