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A236170
Number of partitions of n which have nonconstant reversal sums, as defined in Comments.
2
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
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]] (*A236129*)
Table[PartitionsP[n] - u[[n]], {n, 1, 40}] (*A236170*)
(* 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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 19 2014
STATUS
approved