A262444 Number of 3-colored integer partitions such that no adjacent parts have the same color.

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

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 are 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 * A372559 A109755 A348403

Adjacent sequences: A262441 A262442 A262443 * A262445 A262446 A262447

Keyword

nonn,easy

Author

Ran Pan, Sep 23 2015

Status

approved