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.

1, 2, 5, 10, 20, 40, 72, 127, 217, 362, 587, 954, 1494, 2330, 3562, 5403, 8060, 11954, 17531, 25490, 36733, 52570, 74620, 105273, 147479, 205390, 284516, 391819, 536891, 732028, 993540, 1342174, 1805795, 2419115, 3228530, 4292484, 5686507, 7506642, 9877321

Offset

10,2

Links

Alois P. Heinz, Table of n, a(n) for n = 10..5000

Formula

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

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!)(4):
seq(a(n), n=10..49);

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 = 4}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[10, 49] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A321878.

Sequence in context: A181366 A068034 A222082 * A296122 A293324 A284904

Adjacent sequences: A327284 A327285 A327286 * A327288 A327289 A327290

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2019

Status

approved