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A262444 Number of 3-colored integer partitions so that no adjacent parts have the same color. 2
1, 3, 9, 21, 51, 111, 249, 525, 1119, 2319, 4809, 9825, 20079, 40671, 82341, 165945, 334191, 671307, 1347861, 2702385, 5416395, 10847787, 21720981, 43474869, 87004875, 174081051, 348279777, 696712749, 1393674603, 2787673767, 5575871457, 11152425093, 22305942039 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

FORMULA

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

a(n) = floor(3/2*A070933(n)).

a(n) = Sum_{k=0..3} 6/k! * A262495(n,3-k). - Alois P. Heinz, Sep 24 2015

EXAMPLE

a(2) = 9 because there two integer partitions of 2: [2], [1,1] and there are three ways to color [2] and 3 X 2 = 6 ways to color [1,1].

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))

    end:

a:= n-> floor(b(n$2)/2*3):

seq(a(n), n=0..50);  # Alois P. Heinz, Sep 23 2015

MATHEMATICA

Rest[CoefficientList[Series[3/2 Product[1/(1 - 2 x^k), {k, 1, 35}], {x, 0, 35}], x]] (* Vincenzo Librandi, Sep 23 2015 *)

CROSSREFS

Cf. A070933, A262495.

Sequence in context: A105544 A119917 A111209 * A109755 A005254 A272265

Adjacent sequences:  A262441 A262442 A262443 * A262445 A262446 A262447

KEYWORD

nonn,easy

AUTHOR

Ran Pan, Sep 23 2015

STATUS

approved

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