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A327286 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size three are used and the colors are introduced in increasing order.

%I A327286 #12 Dec 14 2020 07:57:13
%S A327286 1,2,5,10,21,37,67,112,187,302,479,741,1136,1707,2539,3732,5424,7804,
%T A327286 11133,15743,22088,30774,42582,58540,80007,108725,146955,197646,
%U A327286 264525,352433,467541,617651,812734,1065417,1391592,1811296,2349775,3038515,3917052,5034647
%N A327286 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size three are used and the colors are introduced in increasing order.
%H A327286 Alois P. Heinz, <a href="/A327286/b327286.txt">Table of n, a(n) for n = 6..5000</a>
%F A327286 a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-2))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-2)) / (72*Pi*n). - _Vaclav Kotesovec_, Sep 18 2019
%p A327286 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327286      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A327286     end:
%p A327286 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(3):
%p A327286 seq(a(n), n=6..47);
%t A327286 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
%t A327286 a[n_] := With[{k = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
%t A327286 a /@ Range[6, 47] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A327286 Column k=3 of A321878.
%K A327286 nonn
%O A327286 6,2
%A A327286 _Alois P. Heinz_, Aug 28 2019