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%I #26 Mar 23 2020 21:22:14
%S 0,1,2,3,6,11,20,36,64,113,198,345,598,1032,1774,3039,5190,8839,15016,
%T 25452,43052,72685,122502,206133,346346,581136,973850,1630011,2725254,
%U 4551683,7594748,12660660,21087448,35094377,58360134,96979089,161042110,267248664
%N Number of 1's in all compositions of n into odd parts.
%C a(n+1) is the number of ways to tile a strip of length n+1 using white tiles of only odd lengths, with total length n, and one red square of length one. - _Gregory L. Simay_, Aug 14 2016
%C A029907, the number of compositions of n with exactly one even part, is equal to a(n+1-2) + a(n+1-4) + a(n+1-6) + ... - _Gregory L. Simay_, Aug 14 2016
%C Apart from the initial 0 and 1, this is the p-INVERT transform of (1,0,1,0,1,0,...) for p(S) = (1 - S)^2. See A291219. - _Clark Kimberling_, Sep 02 2017
%D S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2010, page 70.
%H Michael De Vlieger, <a href="/A239342/b239342.txt">Table of n, a(n) for n = 0..4771</a>
%H Mengmeng Liu, Andrew Yezhou Wang, <a href="https://www.emis.de/journals/JIS/VOL23/Wang/wang61.html">The Number of Designated Parts in Compositions with Restricted Parts</a>, J. Int. Seq., Vol. 23 (2020), Article 20.1.8.
%F For n >= 4, a(n) = a(n-1) + a(n-2) + A000045(n-2).
%F G.f.: x*(1 - x^2)^2/(1 - x - x^2)^2.
%e a(5) = 11 because in the compositions of 5 into odd parts there are a total of 11 1's: 5, 3+1+1, 1+3+1, 1+1+3, 1+1+1+1+1.
%e Let r represent the red square and 1,3,5 represent the possible odd lengths of the white squares for n=5. Then a(5+1) = a(6) = 20 because r combined with a tile of length 5 generates 2 compositions; r combined with 3,1,1 generates 12 compositions; and r combined with 1,1,1,1,1 generates 6 compositions. 2+12+6 = 20. - _Gregory L. Simay_, Aug 14 2016
%t nn=30;CoefficientList[Series[x (1-x^2)^2/(1-x-x^2)^2,{x,0,nn}],x]
%t (* or *)
%t Table[Count[Flatten[Level[Map[Permutations,IntegerPartitions[n,n,Table[2k+1,{k,0,n/2}]]],{2}]],1],{n,0,30}]
%Y Cf. A000045, A029907.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Mar 16 2014
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