%I #20 Jul 26 2024 04:54:52
%S 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,
%T 60,87,16,148,17,98,86,64,111,242,20,76,119,259,22,308,23,228,341,103,
%U 25,509,146,310,210,343,28,583,359,645,270,151,31,1358,32,169
%N Number of partitions of n that have constant reversal sums; see Comments.
%C 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).
%H Giovanni Resta, <a href="/A236129/b236129.txt">Table of n, a(n) for n = 1..1000</a>
%F 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
%e Of the 11 partitions of 6, the 3 nonconstant reversal sums are listed here: [1,1,4] + [4,1,1] = [5,2,5];
%e [1,1,1,3] + [3,1,1,1] = [4,2,2,4];
%e [1,1,1,1,2] + [2,1,1,1,1] = [3,2,2,2,3].
%e The remaining 8 partitions of 6 have constant reversal sums.
%t u = Map[Length[Select[Map[DeleteDuplicates, Map[Reverse, #] + #], Length[#] == 1 &] &[IntegerPartitions[#]]] &, Range[40]] (*A236129*)
%t Table[PartitionsP[n] - u[[n]], {n, 1, 40}] (*A236170*)
%t (* _Peter J. C. Moses_, Jan 19 2014 *)
%t 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 *)
%Y Cf. A236170.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 19 2014