A236170 Number of partitions of n which have nonconstant reversal sums, as defined in Comments.

0, 0, 0, 1, 3, 3, 10, 13, 20, 29, 49, 54, 93, 116, 151, 206, 287, 342, 479, 579, 752, 968, 1242, 1489, 1923, 2393, 2950, 3631, 4549, 5456, 6825, 8251, 10057, 12246, 14772, 17735, 21617, 25939, 31066, 37079, 44561, 52866, 63238, 74947, 88793, 105455, 124729

Offset

1,5

Comments

A partition [x(1),x(2),...,x(k)] of n has constant reversal sums if x(1) + x(k) = x(2) + x(k-1) = ... = x(k) + x(1). and nonconstant reversal sums otherwise. It is assumed that x(1) <= x(2) <= ... <= x(k).

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

Formula

a(n) = p(n) - sum of C(floor(n/d)+floor(d/2)-1,floor(d/2)) where C(n,k) denotes the binomial coefficient and d ranges among the divisors of 2*n. - Giovanni Resta, Jan 24 2014

Example

Of the 11 partitions of 6, the 3 nonconstant reversal sums are listed here: [1,1,4] + [4,1,1] = [5,2,5];
[1,1,1,3] + [3,1,1,1] = [4,2,2,4];
[1,1,1,1,2] + [2,1,1,1,1] = [3,2,2,2,3].
The remaining 8 partitions of 6 have constant reversal sums.

Mathematica

u = Map[Length[Select[Map[DeleteDuplicates, Map[Reverse, #] + #],    Length[#] == 1 &] &[IntegerPartitions[#]]] &, Range[40]] (*A263129*)
Table[PartitionsP[n] - u[[n]], {n, 1, 40}] (*A263170*)
(* Peter J. C. Moses, Jan 19 2014 *)
a[n_] := PartitionsP[n] - Sum[ Binomial[Floor[n/d] + Floor[d/2] - 1, Floor[d/2]], {d, Divisors[2*n]}] (* Giovanni Resta, Jan 24 2014 *)

Crossrefs

Cf. A236129.

Sequence in context: A218953 A081809 A359894 * A129885 A277963 A193965

Adjacent sequences: A236167 A236168 A236169 * A236171 A236172 A236173

Keyword

nonn,easy

Author

Clark Kimberling, Jan 19 2014

Status

approved