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A236129 Number of partitions of n that have constant reversal sums; see Comments. 2
1, 2, 3, 4, 4, 8, 5, 9, 10, 13, 7, 23, 8, 19, 25, 25, 10, 43, 11, 48, 40, 34, 13, 86, 35, 43, 60, 87, 16, 148, 17, 98, 86, 64, 111, 242, 20, 76, 119, 259, 22, 308, 23, 228, 341, 103, 25, 509, 146, 310, 210, 343, 28, 583, 359, 645, 270, 151, 31, 1358, 32, 169 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A partition [x(1),x(2),...,x(k)], where x(1) <= x(2) <= ... <= x(k), has constant reversal sums if x(1) + x(k) = x(2) + x(k-1) = ... = x(k) + x(1).

LINKS

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

FORMULA

a(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_] := Sum[Binomial[Floor[n/d]+Floor[d/2]-1, Floor[d/2]], {d, Divisors[2 n]}]; Array[a, 62] (* Giovanni Resta, Jan 24 2014 *)

CROSSREFS

Cf. A236170.

Sequence in context: A006087 A241315 A136330 * A240219 A028298 A047966

Adjacent sequences:  A236126 A236127 A236128 * A236130 A236131 A236132

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jan 19 2014

STATUS

approved

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Last modified March 25 20:03 EDT 2017. Contains 284083 sequences.