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A239872 Number of strict partitions of 2n 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 A239872 #11 Oct 28 2015 12:43:41
%S A239872 0,1,1,1,1,1,1,1,2,3,6,10,17,26,40,57,81,110,148,193,250,316,397,491,
%T A239872 603,732,885,1061,1268,1508,1790,2120,2510,2970,3517,4170,4950,5887,
%U A239872 7013,8371,10005,11979,14353,17217,20654,24785,29725,35637,42672,51046,60962
%N A239872 Number of strict partitions of 2n 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 A239872 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 A239873.
%H A239872 Alois P. Heinz, <a href="/A239872/b239872.txt">Table of n, a(n) for n = 0..1000</a>
%e A239872 a(9) counts these 3 partitions of 18:  [18], [8,3,4,1,2], [6,5,4,1,2].
%p A239872 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
%p A239872       abs(t)-n>0, 0, `if`(n=0, 1, b(n, i-1, t)+
%p A239872       `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
%p A239872     end:
%p A239872 a:= n-> b(2*n$2, 1):
%p A239872 seq(a(n), n=0..60);  # _Alois P. Heinz_, Apr 01 2014
%t A239872 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 A239872 TableForm[t] (* shows the partitions *)
%t A239872 u = Table[Length[p[2 n]], {n, 0, 40}]  (* A239872 *)
%t A239872 (* _Peter J. C. Moses_, Mar 10 2014 *)
%t A239872 b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n > 0, 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[2*n, 2*n, 1]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Oct 28 2015, after _Alois P. Heinz_ *)
%Y A239872 Cf. A239241, A239871, A239873, A239832.
%K A239872 nonn,easy
%O A239872 0,9
%A A239872 _Clark Kimberling_, Mar 29 2014