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A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

%I A239873 #12 Aug 29 2016 09:28:01
%S A239873 0,0,0,1,2,4,6,9,12,16,20,25,30,36,43,51,61,74,91,113,144,184,239,311,
%T A239873 407,530,692,895,1155,1478,1882,2375,2983,3715,4602,5660,6925,8418,
%U A239873 10187,12257,14686,17514,20809,24624,29049,34154,40051,46842,54668,63667
%N A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.
%C A239873 Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.  This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239872.
%H A239873 Alois P. Heinz, <a href="/A239873/b239873.txt">Table of n, a(n) for n = 0..1000</a>
%e A239873 a(7) counts these 9 partitions of 15:  [12,1,2], [10,1,4], [10,3,2], [4,9,2], [8,1,6], [8,5,2], [8,3,4], [6,7,2], [6,5,4].
%p A239873 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
%p A239873       abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
%p A239873       `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
%p A239873     end:
%p A239873 a:= n-> b(2*n+1$2, 1):
%p A239873 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 02 2014
%t A239873 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, _?OddQ] == -1 + Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 20}]
%t A239873 TableForm[t] (* shows the partitions *)
%t A239873 u = Table[Length[p[2 n + 1]], {n, 0, 38}]  (* A239873 *)
%t A239873 (* _Peter J. C. Moses_, Mar 10 2014 *)
%t A239873 b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2 Mod[i, 2] - 1)]]]]; a[n_] := b[2n+1, 2n+1, 1]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y A239873 Cf. A239241, A239871, A239872, A239832.
%K A239873 nonn,easy
%O A239873 0,5
%A A239873 _Clark Kimberling_, Mar 29 2014