A239832 - OEIS

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A239832

Number of partitions of n 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.

0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 7, 6, 11, 11, 17, 19, 27, 31, 41, 51, 62, 79, 95, 121, 142, 182, 212, 269, 314, 393, 459, 570, 665, 816, 958, 1160, 1364, 1639, 1928, 2297, 2706, 3200, 3768, 4434, 5212, 6105, 7170, 8361, 9799, 11396, 13322, 15450, 18022

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Let c(n) be the number of partitions of n having 1 more odd part than even, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are odd. Then c(n) = a(n+1) for n >= 0.

LINKS

Table of n, a(n) for n=0..53.

EXAMPLE

The three partitions counted by a(10) are [10], [4,1,2,1,2], and [2,3,2,1,2].

MATHEMATICA

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, _?OddQ] == -1 + Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 10}]

TableForm[t] (* shows the partitions *)

Table[Length[p[n]], {n, 0, 30}] (* A239832 *)

(* Peter J. C. Moses, Mar 10 2014 *)

CROSSREFS

Cf. A239833, A239835, A045931, A239871.

Column k=-1 of A240009.