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A262445
Number of exact 3-colored partitions such that no adjacent parts have the same color.
3
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
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
KEYWORD
nonn
AUTHOR
Ran Pan, Sep 23 2015
STATUS
approved