

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.


4



0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 43, 51, 61, 74, 91, 113, 144, 184, 239, 311, 407, 530, 692, 895, 1155, 1478, 1882, 2375, 2983, 3715, 4602, 5660, 6925, 8418, 10187, 12257, 14686, 17514, 20809, 24624, 29049, 34154, 40051, 46842, 54668, 63667
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

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.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000


EXAMPLE

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].


MAPLE

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
abs(t)>n, 0, `if`(n=0, 1, b(n, i1, t)+
`if`(i>n, 0, b(ni, i1, t+(2*irem(i, 2)1)))))
end:
a:= n> b(2*n+1$2, 1):
seq(a(n), n=0..80); # Alois P. Heinz, Apr 02 2014


MATHEMATICA

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}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n + 1]], {n, 0, 38}] (* A239873 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2  Abs[t] > n, 0, If[n == 0, 1, b[n, i1, t] + If[i>n, 0, b[ni, i1, t + (2 Mod[i, 2]  1)]]]]; a[n_] := b[2n+1, 2n+1, 1]; Table[a[n], {n, 0, 80}] (* JeanFrançois Alcover, Aug 29 2016, after Alois P. Heinz *)


CROSSREFS

Cf. A239241, A239871, A239872, A239832.
Sequence in context: A224812 A194256 A194246 * A224811 A024617 A025698
Adjacent sequences: A239870 A239871 A239872 * A239874 A239875 A239876


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Mar 29 2014


STATUS

approved



