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A242499 Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 1.

%I A242499 #14 Dec 28 2020 04:23:54
%S A242499 1,0,1,3,1,9,11,18,51,65,151,290,477,1043,1835,3486,6931,12540,24607,
%T A242499 46797,87979,171072,323269,619245,1190619,2264925,4357211,8343322,
%U A242499 15973309,30711853,58846191,113027716,217192103,416964202,801880039,1541412015,2963997227
%N A242499 Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 1.
%C A242499 With offset 2 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -1.
%H A242499 Alois P. Heinz, <a href="/A242499/b242499.txt">Table of n, a(n) for n = 1..1000</a>
%F A242499 Recurrence (for n>=5): (n+2)*(16*n^4 - 128*n^3 + 344*n^2 - 352*n + 89)*a(n) = -32*(n+1)*(2*n-5)*a(n-1) + 2*(16*n^5 - 112*n^4 + 264*n^3 - 320*n^2 + 301*n - 89)*a(n-2) + 2*(2*n-5)*(16*n^4 - 80*n^3 + 80*n^2 + 36*n - 53)*a(n-3) - (n-4)*(16*n^4 - 64*n^3 + 56*n^2 + 16*n - 31)*a(n-4). - _Vaclav Kotesovec_, May 20 2014
%p A242499 a:= proc(n) option remember;
%p A242499       `if`(n<6, [0, 1, 0, 1, 3, 1][n+1],
%p A242499       ((3*n-2)*a(n-2) +(4*n+2)*a(n-3) -(3*n-10)*a(n-4)
%p A242499        -(4*n-22)*a(n-5) +(n-6)*a(n-6))/(n+2))
%p A242499     end:
%p A242499 seq(a(n), n=1..50);
%t A242499 a[n_] := a[n] = If[n<6, {0, 1, 0, 1, 3, 1}[[n+1]], ((3n-2)a[n-2] + (4n+2)a[n-3] - (3n-10)a[n-4] - (4n-22)a[n-5] + (n-6)a[n-6])/(n+2)];
%t A242499 Array[a, 50] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A242499 Column k=1 of A242498.
%K A242499 nonn
%O A242499 1,4
%A A242499 _Alois P. Heinz_, May 16 2014