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%I #12 Aug 29 2016 09:28:01
%S 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 407,530,692,895,1155,1478,1882,2375,2983,3715,4602,5660,6925,8418,
%U 10187,12257,14686,17514,20809,24624,29049,34154,40051,46842,54668,63667
%N 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 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 Alois P. Heinz, <a href="/A239873/b239873.txt">Table of n, a(n) for n = 0..1000</a>
%e 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 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
%p abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
%p `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
%p end:
%p a:= n-> b(2*n+1$2, 1):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Apr 02 2014
%t 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 TableForm[t] (* shows the partitions *)
%t u = Table[Length[p[2 n + 1]], {n, 0, 38}] (* A239873 *)
%t (* _Peter J. C. Moses_, Mar 10 2014 *)
%t 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 Cf. A239241, A239871, A239872, A239832.
%K nonn,easy
%O 0,5
%A _Clark Kimberling_, Mar 29 2014
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