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A327381 Number of colored integer partitions of n such that three colors are used and parts differ by size or by color. 6
1, 3, 9, 19, 39, 72, 128, 216, 354, 563, 876, 1335, 1998, 2946, 4284, 6154, 8742, 12294, 17129, 23667, 32442, 44151, 59682, 80169, 107054, 142167, 187812, 246895, 323058, 420852, 545958, 705438, 908043, 1164609, 1488504, 1896174, 2407836, 3048255, 3847716 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

With offset 0 convolution cube of A000009(k+1). - George Beck, Jan 29 2021

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 3..10000 (terms 3..5000 from Alois P. Heinz)

Wikipedia, Partition (number theory)

FORMULA

a(n) ~ exp(Pi*sqrt(n)) / (8 * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^3. - 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))(3):

seq(a(n), n=3..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 = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];

a /@ Range[3, 45] (* Jean-Fran├žois Alcover, Dec 15 2020, after Alois P. Heinz *)

CROSSREFS

Column k=3 of A308680.

Cf. A000009.

Sequence in context: A080010 A135117 A038163 * A146819 A147213 A146441

Adjacent sequences:  A327378 A327379 A327380 * A327382 A327383 A327384

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 03 2019

STATUS

approved

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Last modified June 26 05:02 EDT 2022. Contains 354877 sequences. (Running on oeis4.)