A239880 Number of strict partitions of n having an ordering in which no parts of equal parity are juxtaposed and the first and last terms have the same parity.

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 11, 14, 15, 19, 22, 26, 30, 35, 42, 47, 56, 62, 76, 83, 100, 108, 132, 142, 171, 184, 222, 239, 284, 306, 363, 394, 460, 500, 581, 636, 730, 802, 914, 1010, 1139, 1262, 1415, 1577, 1753, 1956, 2163, 2423, 2663

Offset

0,7

Comments

A strict partition is one in which every part has multiplicity 1.

a(n) = A240021(n,-1) + A240021(n,1). - Alois P. Heinz, Apr 02 2014

Links

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

Example

a(12) counts these 7 partitions:  [12], [9,2,1], [3,8,1], [7,4,1], [7,2,3], [5,6,1], [5,4,3].

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
       abs(t)-n>1, 0, `if`(n=0, abs(t), b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
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], Abs[Count[#, _?OddQ] - Count[#, _?EvenQ]] == 1 &]; t =  Table[p[n], {n, 0, 12}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[n]], {n, 0, 60}]  (* A239880 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n>1, 0, If[n==0, Abs[t], 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, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Crossrefs

Cf. A239833, A239835, A239881.

Sequence in context: A040039 A008667 A367221 * A240862 A177716 A109763

Adjacent sequences: A239877 A239878 A239879 * A239881 A239882 A239883

Keyword

nonn,easy

Author

Clark Kimberling, Mar 29 2014

Status

approved