A242110 Number of partitions of n whose different summands alternate in parity.

1, 1, 2, 3, 4, 6, 8, 11, 13, 21, 23, 33, 39, 54, 63, 88, 98, 132, 157, 200, 237, 303, 356, 440, 526, 643, 767, 931, 1103, 1317, 1581, 1860, 2215, 2615, 3100, 3631, 4302, 4999, 5907, 6865, 8059, 9322, 10950, 12613, 14744, 16988, 19756, 22694, 26344, 30192

Offset

0,3

Comments

If the largest part is even (odd ), then the second largest part must be odd (even), the third largest part even (odd),...

Links

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

Shane Chern, Unlimited parity alternating partitions, arXiv:1803.01031 [math.CO], 2018.

Example

The first of the unrestricted partitions not to be counted is 3+1, because the largest part, 3, is odd and the next largest part, 1, is also odd.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, t) +`if`(irem(i+t, 2)=0, 0,
      add(b(n-i*j, i-1, 1-t), j=1..n/i))))
    end:
a:= n-> `if`(n=0, 1, add(b(n$2, j), j=0..1)):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 17 2014

Mathematica

<<Combinatorica`;
For[n=1, n<=20, n++, count[n]=1;
p={n};
For[index=1, index<=PartitionsP[n]-1, index++,
p=NextPartition[p];
condition=True;
For[i=1, i<=Length[p]-1, i++,
If[((p[[i]]!=p[[i+1]])&&EvenQ[p[[i]]]&&EvenQ[p[[i+1]]])||
((p[[i]]!=p[[i+1]]&&OddQ[p[[i]]])&&OddQ[p[[i+1]]]), condition=False]];
If[condition, count[n]++]];
];
Print[Table[count[i], {i, 1, n-1}]]

Crossrefs

Sequence in context: A347462 A340589 A102463 * A371069 A056829 A211536

Adjacent sequences: A242107 A242108 A242109 * A242111 A242112 A242113

Keyword

nonn

Author

David S. Newman, Aug 15 2014

Extensions

More terms from Alois P. Heinz, Aug 17 2014

Status

approved