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A327380
Number of colored integer partitions of n such that two colors are used and parts differ by size or by color.
5
1, 2, 5, 8, 14, 22, 34, 50, 73, 104, 146, 202, 275, 372, 498, 660, 868, 1134, 1470, 1896, 2430, 3098, 3931, 4964, 6240, 7814, 9746, 12110, 14997, 18510, 22772, 27934, 34166, 41672, 50698, 61520, 74470, 89940, 108378, 130312, 156364, 187244, 223785, 266962
OFFSET
2,2
COMMENTS
With offset 0 convolution square of A000009(k+1). - George Beck, Jan 28 2021
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 2..10000 (terms 2..5000 from Alois P. Heinz)
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019
G.f.: (-1 + Product_{j>=1} (1 + x^j))^2. - Alois P. Heinz, Jan 29 2021
EXAMPLE
a(4) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
a(5) = 8: 2a2b1a, 2a2b1b, 3a1a1b, 3b1a1b, 3a2b, 3b2a, 4a1b, 4b1a.
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))(2):
seq(a(n), n=2..45);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 2}, Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 45] (* Jean-François Alcover, May 06 2020, after Maple *)
CROSSREFS
Column k=2 of A308680.
Cf. A000009.
Sequence in context: A011842 A000094 A182377 * A330378 A058578 A261526
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 03 2019
STATUS
approved