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A327288 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size five are used and the colors are introduced in increasing order.

%I A327288 #10 Dec 14 2020 09:10:32
%S A327288 1,2,5,10,20,36,73,125,222,372,623,1002,1611,2559,3984,6139,9355,
%T A327288 14096,21028,31093,45523,66403,95779,137495,195813,277531,390428,
%U A327288 546942,761113,1054749,1454412,1996271,2727247,3711683,5029288,6789347,9130315,12234596,16335987
%N A327288 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size five are used and the colors are introduced in increasing order.
%H A327288 Alois P. Heinz, <a href="/A327288/b327288.txt">Table of n, a(n) for n = 15..5000</a>
%F A327288 a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-4))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-4)) / (4*5!*sqrt(15)*Pi*n). - _Vaclav Kotesovec_, Sep 18 2019
%p A327288 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327288      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A327288     end:
%p A327288 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(5):
%p A327288 seq(a(n), n=15..53);
%t A327288 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 A327288 a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
%t A327288 a /@ Range[15, 53] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A327288 Column k=5 of A321878.
%K A327288 nonn
%O A327288 15,2
%A A327288 _Alois P. Heinz_, Aug 28 2019