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%I #36 Nov 04 2017 16:15:45
%S 1,2,4,8,16,26,50,80,130,212,342,518,820,1276,1864,2960,4336,6704,
%T 9710,15068,21368,33420,47082,72950,102316,158888,220882,342616,
%U 475108,734816,1015778,1569680,2161944,3337952,4587200,7069748,9699292,14932444,20445520
%N a(n) is the total number of k-reverses of n.
%C See sequence A180171 for the definition of a k-reverse of n.
%C Briefly, a k-reverse of n is a k-composition of n whose reverse is cyclically equivalent to itself.
%C This sequence is the total number of k-reverses of n for k=1,2,...,n.
%C It is the row sums of the 'R(n,k)' triangle from sequence A180171.
%C For example a(6)=26 because there are 26 k-reverses of n=6 for k=1,2,3,4,5, or 6.
%C They are, in cyclically equivalent, classes: {6}, {15,51}, {24,42},{33},{114,411,141},{222} {1113,3111,1311,1131}, {1122,2112,2211,1221}, {1212,2121}, {11112,21111,12111,11211,11121}, {111111}.
%D John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010.
%H Andrew Howroyd, <a href="/A180249/b180249.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = Sum_{d|n} d*A056493(d)/2. - _Andrew Howroyd_, Oct 07 2017
%F From _Petros Hadjicostas_, Oct 15 2017: (Start)
%F a(n) = (n/2)*Sum_{d|n} (phi^(-1)(d)/d)*b(n/d), where phi^(-1)(n) = A023900(n) is the Dirichlet inverse of the Euler totient function and b(n) = A029744(n+1) (= 3*2^((n/2)-1), if n is even, and = 2^((n+1)/2), if n is odd).
%F G.f.: Sum_{n>=1} phi^(-1)(n)*g(x^n), where phi^(-1)(n) = A023900(n) and g(x) = x*(x+1)*(2*x+1)/(1-2*x^2)^2.
%F (End)
%t f[n_Integer] := Block[{c = 0, k = 1, ip = IntegerPartitions@ n, lmt = 1 + PartitionsP@ n, ipk}, While[k < lmt, c += g[ ip[[k]]]; k++ ]; c]; g[lst_List] := Block[{c = 0, len = Length@ lst, per = Permutations@ lst}, While[ Length@ per > 0, rl = Union[ RotateLeft[ per[[1]], # ] & /@ Range@ len]; If[ MemberQ[rl, Reverse@ per[[1]]], c += Length@ rl]; per = Complement[ per, rl]]; c]; Array[f, 24] (* _Robert G. Wilson v_, Aug 25 2010 *)
%t b[n_] := Sum[MoebiusMu[n/d] * If[OddQ[d], 2, 3] * 2^Quotient[d-1, 2], {d, Divisors[n]}]; a[n_] := Sum[d*b[d], {d, Divisors[n]}] / 2; Array[a, 39] (* _Jean-François Alcover_, Nov 04 2017, after _Andrew Howroyd_ *)
%o (PARI) \\ here b(n) is A056493
%o b(n) = sumdiv(n, d, moebius(n/d) * if(d%2,2,3) * 2^((d-1)\2));
%o a(n) = sumdiv(n, d, d*b(d)) / 2; \\ _Andrew Howroyd_, Oct 07 2017
%Y If we ask for the number of cyclically equivalent classes we get sequence A052955.
%Y For example the 6th term of A052955 is 11, corresponding to the 11 classes in the example above.
%Y Row sums of A180171.
%Y Cf. A056493, A180322.
%K nonn
%O 1,2
%A _John P. McSorley_, Aug 19 2010
%E a(11) - a(24) from _Robert G. Wilson v_, Aug 25 2010
%E a(25) - a(27) from _Robert G. Wilson v_, Aug 29 2010
%E Terms a(28) and beyond from _Andrew Howroyd_, Oct 07 2017
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