A262445 Number of exact 3-colored partitions such that no adjacent parts have the same color.

0, 0, 0, 6, 24, 72, 186, 438, 990, 2142, 4560, 9492, 19620, 40068, 81534, 164892, 332808, 669528, 1345554, 2699448, 5412636, 10843038, 21714972, 43467342, 86995428, 174069306, 348265164, 696694692, 1393652298, 2787646380, 5575837836, 11152384044, 22305891948, 44613248352, 89228806704, 178460625402, 356925987924

Offset

0,4

Comments

a(1) = a(2) = 0 because we need to use exactly three colors, which means the number of parts should be greater than two.

All terms are multiples of 6.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Ran Pan, A note on enumerating colored integer partitions, arXiv:1509.06107 [math.CO], 2015.

Ran Pan, Exercise S, Project P.

Formula

G.f.: 3/2*Product_{k>=1} (1/(1-2*x^k)) - 6*Product_{k>=1} (1/(1-x^k)) + 3/(1-x) + 3/2.

a(n) = A262444(n) - 6*A000041(n) + 3, for n >= 1.

a(n) = 6 * A262495(n,3). - Alois P. Heinz, Sep 24 2015

Example

a(3)=6 because there are three partitions of 3 and there are no ways to color [3] or [2,1] but there are six ways to color [1,1,1].

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 0, b(n$2, 2)/2*3-6*b(n$2, 1)+3):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 23 2015

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := If[n == 0, 0, b[n, n, 2]/2*3 - 6*b[n, n, 1] + 3]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A262444, A139582, A262495, A320545.

Sequence in context: A006528 A052749 A337021 * A090574 A294842 A290132

Adjacent sequences: A262442 A262443 A262444 * A262446 A262447 A262448

Keyword

nonn

Author

Ran Pan, Sep 23 2015

Status

approved