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A327383
Number of colored integer partitions of n such that five colors are used and parts differ by size or by color.
6
1, 5, 20, 60, 160, 381, 845, 1760, 3495, 6660, 12267, 21935, 38230, 65140, 108785, 178437, 287975, 457965, 718575, 1113680, 1706533, 2587655, 3885615, 5781830, 8530625, 12486429, 18140360, 26169335, 37501595, 53403915, 75597130, 106408670, 148973260, 207496090
OFFSET
5,2
COMMENTS
With offset 0 five-fold convolution of A000009(k+1). - George Beck, Jan 29 2021
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 5..10000 (terms 5..5000 from Alois P. Heinz)
FORMULA
a(n) ~ exp(Pi*sqrt(5*n/3)) * 5^(1/4) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019
G.f.: (-1 + Product_{m >= 1} (1 + x^m))^5. - George Beck, Jan 29 2021
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)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):
seq(a(n), n=5..45);
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] Binomial[k, j]], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[5, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)
CROSSREFS
Column k=5 of A308680.
Cf. A000009.
Sequence in context: A319888 A319869 A038165 * A339588 A344099 A215224
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 03 2019
STATUS
approved