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

Author

Ran Pan, Sep 23 2015

Keywords

• nonn
• easy

Examples

			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].
		

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.

Crossrefs

Cf. A070933, A262495.

Programs

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 *)

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