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A262445 Number of exact 3-colored partitions so that no adjacent parts have the same color. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 07 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A262444, A139582, A262495.

Sequence in context: A234271 A006528 A052749 * A090574 A290132 A225383

Adjacent sequences:  A262442 A262443 A262444 * A262446 A262447 A262448

KEYWORD

nonn

AUTHOR

Ran Pan, Sep 23 2015

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.