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A327288
Number of partitions of n into colored blocks of equal parts, such that all colors from a set of size five are used and the colors are introduced in increasing order.
2
1, 2, 5, 10, 20, 36, 73, 125, 222, 372, 623, 1002, 1611, 2559, 3984, 6139, 9355, 14096, 21028, 31093, 45523, 66403, 95779, 137495, 195813, 277531, 390428, 546942, 761113, 1054749, 1454412, 1996271, 2727247, 3711683, 5029288, 6789347, 9130315, 12234596, 16335987
OFFSET
15,2
LINKS
FORMULA
a(n) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2,-4))*n/3)) * sqrt(Pi^2 - 6*polylog(2,-4)) / (4*5!*sqrt(15)*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!)(5):
seq(a(n), n=15..53);
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 = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]/ k!];
a /@ Range[15, 53] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)
CROSSREFS
Column k=5 of A321878.
Sequence in context: A327290 A227356 A327289 * A102688 A236559 A275388
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 28 2019
STATUS
approved