cp's OEIS Frontend

A239871 Number of strict partitions of n 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 A239871 #13 Nov 16 2015 08:06:16
%S A239871 0,0,1,0,1,0,1,1,1,2,1,4,1,6,1,9,2,12,3,16,6,20,10,25,17,30,26,36,40,
%T A239871 43,57,51,81,61,110,74,148,91,193,113,250,144,316,184,397,239,491,311,
%U A239871 603,407,732,530,885,692,1061,895,1268,1155,1508,1478,1790
%N A239871 Number of strict partitions of n 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 A239871 Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of n having 1 more odd part than even, so that there is an ordering of parts for which the odd and even parts alternate and the first and last terms are odd.  Then c(n) = a(n+1) for n >= 0.
%H A239871 Alois P. Heinz, <a href="/A239871/b239871.txt">Table of n, a(n) for n = 0..1000</a>
%F A239871 a(n) = [x^n y^(-1)] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1). - _Alois P. Heinz_, Apr 03 2014
%e A239871 a(11) counts these 4 partitions:  812, 614, 632, 452.
%p A239871 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
%p A239871       abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
%p A239871       `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
%p A239871     end:
%p A239871 a:= n-> b(n$2, 1):
%p A239871 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 02 2014
%t A239871 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A239871 p[n_] := p[n] = Select[d[n], Count[#, _?OddQ] == -1 + Count[#, _?EvenQ] &]; t =  Table[p[n], {n, 0, 20}]
%t A239871 TableForm[t] (* shows the partitions *)
%t A239871 u = Table[Length[p[n]], {n, 0, 70}]  (* A239871 *)
%t A239871 (* _Peter J. C. Moses_, Mar 10 2014 *)
%t A239871 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[n, n, 1]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 16 2015, after _Alois P. Heinz_ *)
%Y A239871 Cf. A239241, A239872, A239873, A239832.
%Y A239871 Column k=-1 of A240021.
%K A239871 nonn,easy
%O A239871 0,10
%A A239871 _Clark Kimberling_, Mar 29 2014