

A180249


a(n) is the total number of kreverses of n.


4



1, 2, 4, 8, 16, 26, 50, 80, 130, 212, 342, 518, 820, 1276, 1864, 2960, 4336, 6704, 9710, 15068, 21368, 33420, 47082, 72950, 102316, 158888, 220882, 342616, 475108, 734816, 1015778, 1569680, 2161944, 3337952, 4587200, 7069748, 9699292, 14932444, 20445520
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OFFSET

1,2


COMMENTS

See sequence A180171 for the definition of a kreverse of n.
Briefly, a kreverse of n is a kcomposition of n whose reverse is cyclically equivalent to itself.
This sequence is the total number of kreverses of n for k=1,2,...,n.
It is the row sums of the 'R(n,k)' triangle from sequence A180171.
For example a(6)=26 because there are 26 kreverses of n=6 for k=1,2,3,4,5, or 6.
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}.


REFERENCES

John P. McSorley: Counting kcompositions with palindromic and related structures. Preprint, 2010.


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200


FORMULA

a(n) = Sum_{dn} d*A056493(d)/2.  Andrew Howroyd, Oct 07 2017
From Petros Hadjicostas, Oct 15 2017: (Start)
a(n) = (n/2)*Sum_{dn} (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).
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)/(12*x^2)^2.
(End)


MATHEMATICA

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 *)
b[n_] := Sum[MoebiusMu[n/d] * If[OddQ[d], 2, 3] * 2^Quotient[d1, 2], {d, Divisors[n]}]; a[n_] := Sum[d*b[d], {d, Divisors[n]}] / 2; Array[a, 39] (* JeanFrançois Alcover, Nov 04 2017, after Andrew Howroyd *)


PROG

(PARI) \\ here b(n) is A056493
b(n) = sumdiv(n, d, moebius(n/d) * if(d%2, 2, 3) * 2^((d1)\2));
a(n) = sumdiv(n, d, d*b(d)) / 2; \\ Andrew Howroyd, Oct 07 2017


CROSSREFS

If we ask for the number of cyclically equivalent classes we get sequence A052955.
For example the 6th term of A052955 is 11, corresponding to the 11 classes in the example above.
Row sums of A180171.
Cf. A056493, A180322.
Sequence in context: A354146 A354255 A319385 * A060957 A322326 A018826
Adjacent sequences: A180246 A180247 A180248 * A180250 A180251 A180252


KEYWORD

nonn


AUTHOR

John P. McSorley, Aug 19 2010


EXTENSIONS

a(11)  a(24) from Robert G. Wilson v, Aug 25 2010
a(25)  a(27) from Robert G. Wilson v, Aug 29 2010
Terms a(28) and beyond from Andrew Howroyd, Oct 07 2017


STATUS

approved



