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.

1, 2, 5, 9, 17, 28, 47, 74, 116, 175, 263, 385, 560, 800, 1135, 1589, 2210, 3041, 4160, 5642, 7609, 10189, 13575, 17976, 23694, 31066, 40559, 52708, 68230, 87957, 112985, 144594, 184437, 234466, 297159, 375453, 473039, 594298, 744681, 930674, 1160271, 1442989

Offset

3,2

Links

Alois P. Heinz, Table of n, a(n) for n = 3..5000

Formula

a(n) ~ exp(Pi*sqrt(n)) / (16*n). - Vaclav Kotesovec, Sep 18 2019

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!)(2):
seq(a(n), n=3..44);

Mathematica

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]]];
a[n_] := With[{k = 2}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[3, 44] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Crossrefs

Column k=2 of A321878.

Sequence in context: A366738 A068006 A000097 * A081996 A034329 A230441

Adjacent sequences: A327282 A327283 A327284 * A327286 A327287 A327288

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2019

Status

approved