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%I #30 Sep 02 2023 18:48:16
%S 1,3,9,21,51,111,249,525,1119,2319,4809,9825,20079,40671,82341,165945,
%T 334191,671307,1347861,2702385,5416395,10847787,21720981,43474869,
%U 87004875,174081051,348279777,696712749,1393674603,2787673767,5575871457,11152425093,22305942039
%N Number of 3-colored integer partitions such that no adjacent parts have the same color.
%H Alois P. Heinz, <a href="/A262444/b262444.txt">Table of n, a(n) for n = 0..1000</a>
%H Ran Pan, <a href="http://arxiv.org/abs/1509.06107">A note on enumerating colored integer partitions</a>, arXiv:1509.06107 [math.CO], 2015.
%F G.f.: -1/2 + (3/2)*Product_{k>=1} 1/(1-2*x^k).
%F a(n) = floor(3/2*A070933(n)).
%F a(n) = Sum_{k=0..3} 6/k! * A262495(n,3-k). - _Alois P. Heinz_, Sep 24 2015
%e 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].
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
%p end:
%p a:= n-> floor(b(n$2)/2*3):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Sep 23 2015
%t Rest[CoefficientList[Series[3/2 Product[1/(1 - 2 x^k), {k, 1, 35}], {x, 0, 35}], x]] (* _Vincenzo Librandi_, Sep 23 2015 *)
%Y Cf. A070933, A262495.
%K nonn,easy
%O 0,2
%A _Ran Pan_, Sep 23 2015
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