OFFSET
0,4
COMMENTS
Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent, as in Example.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k=-1..1} A240009(n,k). - Alois P. Heinz, Apr 01 2014
EXAMPLE
a(8) counts these 8 partitions: 8, 161, 521, 341, 4121, 323, 3212, 21212.
MAPLE
b:= proc(n, i, t) option remember; `if`(abs(t)-n>1, 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, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Apr 01 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 *)
Table[Length[p[n]], {n, 0, 60}] (* A239835 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]-n>1, 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, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 29 2014
STATUS
approved