A239833 Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent and the first and last terms have the same parity.

0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 17, 22, 28, 36, 46, 58, 72, 92, 113, 141, 174, 216, 263, 324, 394, 481, 583, 707, 852, 1029, 1235, 1481, 1774, 2118, 2524, 3003, 3567, 4225, 5003, 5906, 6968, 8202, 9646, 11317, 13275, 15531, 18160, 21195, 24718, 28772

Offset: 0

Author

Clark Kimberling, Mar 29 2014

Keywords

• nonn
• easy

Examples

			a(10) counts these 10 partitions:  [10], [1,8,1], [7,2,1], [3,6,1], [5,4,1], [5,3,2], [3,4,3], [4,1,2,1,2], [2,3,2,1,2], [1,2,1,2,1,2,1].
		

Links

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

Crossrefs

Cf. A239832, A239835, A045931, A239871.

Programs

Maple

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

Mathematica

p[n_] := p[n] = Select[IntegerPartitions[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] == 1 &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions*)
t = Table[Length[p[n]], {n, 0, 60}] (* A239833 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]>n, 0, If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]]; a[n_] := b[n, n, -1] + b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

Formula

a(n) = A239832(n) + A239832(n+1) for n >= 0.

a(n) = A240009(n,-1) + A240009(n,1). - Alois P. Heinz, Apr 02 2014