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

Author

Ran Pan, Sep 23 2015

Keywords

• nonn

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.

Examples

			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].
		

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.

Crossrefs

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

Programs

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 *)

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