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A331606 Number of compositions of n with the multiplicity of the first part odd.

%I A331606 #29 May 06 2022 13:13:51
%S A331606 1,1,4,4,12,18,44,72,158,288,604,1146,2332,4528,9126,17944,35940,
%T A331606 71130,142132,282344,563630,1121936,2239060,4462530,8906236,17764160,
%U A331606 35458774,70761520,141272876,282025466,563159588,1124543256,2245918406,4485670168,8960061076
%N A331606 Number of compositions of n with the multiplicity of the first part odd.
%H A331606 Alois P. Heinz, <a href="/A331606/b331606.txt">Table of n, a(n) for n = 1..1000</a>
%H A331606 M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Archibald/arch3.html">Inversions and Parity in Compositions of Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
%F A331606 G.f.: Sum_{i>=1} (1-x)*x^i/(2*(-2*x^(i+1)+2*x^i-2*x+1)) + x/(2*(1-2*x)).
%F A331606 a(n) = A011782(n) - A331609(n). - _Alois P. Heinz_, Jan 23 2020
%e A331606 For n=3, a(4)=4 as we count 4, 3+1, 1+3 and 2+1+1.
%p A331606 b:= proc(n, p, t) option remember; `if`(n=0, t,
%p A331606       add(b(n-j, p, `if`(p=j, 1-t, t)), j=1..n))
%p A331606     end:
%p A331606 a:= n-> add(b(n-j, j, 1), j=1..n):
%p A331606 seq(a(n), n=1..38);  # _Alois P. Heinz_, Jan 23 2020
%t A331606 gf[x_] := x/(2 (1 - 2 x)) + Sum[(1 - x) x^i/(2 (-2 x^(i + 1) + 2 x^i - 2 x + 1))  , {i, 1, 40}]; CL := CoefficientList[Series[gf[x], {x, 0, 35}], x];
%t A331606 Drop[CL, 1] (* _Peter Luschny_, Jan 23 2020 *)
%Y A331606 Cf. A011782, A331609 (similar, with even).
%K A331606 nonn
%O A331606 1,3
%A A331606 _Arnold Knopfmacher_, Jan 22 2020