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Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.
11

%I #8 Feb 13 2020 11:50:49

%S 1,0,0,2,2,4,4,6,6,8,32,34,58,84,132,158,230,280,376,450,570,1388,

%T 1556,2398,3310,4920,6600,9674,12122,16684,21340,28110,34974,45392,

%U 55208,69274,124498,143676,204012,270758,377966,493024,690304,895434,1223826,1562948

%N Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.

%H Alois P. Heinz, <a href="/A332305/b332305.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: Sum_{k>=0} (2*k)! * x^(k*(2*k + 1)) / Product_{j=1..2*k} (1 - x^j).

%F a(n) = A032020(n) - A332304(n).

%e a(5) = 4 because we have [4, 1], [3, 2], [2, 3] and [1, 4].

%p b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,

%p irem(p+1, 2)*p!, add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Feb 09 2020

%t nmax = 45; CoefficientList[Series[Sum[(2 k)! x^(k (2 k + 1))/Product[1 - x^j, {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A027187, A032020, A034008, A067661, A332304.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Feb 09 2020