A327286 - OEIS

%I #12 Dec 14 2020 07:57:13

%S 1,2,5,10,21,37,67,112,187,302,479,741,1136,1707,2539,3732,5424,7804,

%T 11133,15743,22088,30774,42582,58540,80007,108725,146955,197646,

%U 264525,352433,467541,617651,812734,1065417,1391592,1811296,2349775,3038515,3917052,5034647

%N 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 Alois P. Heinz, <a href="/A327286/b327286.txt">Table of n, a(n) for n = 6..5000</a>

%F 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 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))

%p     end:

%p a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(3):

%p seq(a(n), n=6..47);

%t 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 a[n_] := With[{k = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];

%t a /@ Range[6, 47] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)

%Y Column k=3 of A321878.

%K nonn

%O 6,2

%A _Alois P. Heinz_, Aug 28 2019