cp's OEIS Frontend

A327287 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size four are used and the colors are introduced in increasing order.

%I A327287 #10 Dec 14 2020 08:52:20
%S A327287 1,2,5,10,20,40,72,127,217,362,587,954,1494,2330,3562,5403,8060,11954,
%T A327287 17531,25490,36733,52570,74620,105273,147479,205390,284516,391819,
%U A327287 536891,732028,993540,1342174,1805795,2419115,3228530,4292484,5686507,7506642,9877321
%N A327287 Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size four are used and the colors are introduced in increasing order.
%H A327287 Alois P. Heinz, <a href="/A327287/b327287.txt">Table of n, a(n) for n = 10..5000</a>
%F A327287 a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-3))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-3)) / (4*4!*sqrt(12)*Pi*n). - _Vaclav Kotesovec_, Sep 18 2019
%p A327287 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327287      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
%p A327287     end:
%p A327287 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(4):
%p A327287 seq(a(n), n=10..49);
%t A327287 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 A327287 a[n_] := With[{k = 4}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
%t A327287 a /@ Range[10, 49] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A327287 Column k=4 of A321878.
%K A327287 nonn
%O A327287 10,2
%A A327287 _Alois P. Heinz_, Aug 28 2019