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A327285 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size two are used and the colors are introduced in increasing order.

%I A327285 #12 Dec 14 2020 08:44:29
%S A327285 1,2,5,9,17,28,47,74,116,175,263,385,560,800,1135,1589,2210,3041,4160,
%T A327285 5642,7609,10189,13575,17976,23694,31066,40559,52708,68230,87957,
%U A327285 112985,144594,184437,234466,297159,375453,473039,594298,744681,930674,1160271,1442989
%N A327285 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size two are used and the colors are introduced in increasing order.
%H A327285 Alois P. Heinz, <a href="/A327285/b327285.txt">Table of n, a(n) for n = 3..5000</a>
%F A327285 a(n) ~ exp(Pi*sqrt(n)) / (16*n). - _Vaclav Kotesovec_, Sep 18 2019
%p A327285 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327285      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A327285     end:
%p A327285 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(2):
%p A327285 seq(a(n), n=3..44);
%t A327285 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 A327285 a[n_] := With[{k = 2}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
%t A327285 a /@ Range[3, 44] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A327285 Column k=2 of A321878.
%K A327285 nonn
%O A327285 3,2
%A A327285 _Alois P. Heinz_, Aug 28 2019