A239872 Number of strict partitions of 2n 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.

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 10, 17, 26, 40, 57, 81, 110, 148, 193, 250, 316, 397, 491, 603, 732, 885, 1061, 1268, 1508, 1790, 2120, 2510, 2970, 3517, 4170, 4950, 5887, 7013, 8371, 10005, 11979, 14353, 17217, 20654, 24785, 29725, 35637, 42672, 51046, 60962

Offset

0,9

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 A239873.

Links

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

Example

a(9) counts these 3 partitions of 18:  [18], [8,3,4,1,2], [6,5,4,1,2].

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)-n>0, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(2*n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 01 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]], {n, 0, 40}]  (* A239872 *)
(* 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, 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[2*n, 2*n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

Crossrefs

Cf. A239241, A239871, A239873, A239832.

Sequence in context: A347786 A036588 A334893 * A099517 A026647 A026669

Adjacent sequences: A239869 A239870 A239871 * A239873 A239874 A239875

Keyword

nonn,easy

Author

Clark Kimberling, Mar 29 2014

Status

approved