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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.
2
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 28 2019
STATUS
approved