This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179968 #28 Aug 23 2022 10:19:41
%S A179968 1,1,0,1,2,0,1,2,3,0,1,4,6,4,0,1,4,9,8,5,0,1,6,15,20,15,6,0,1,6,21,32,
%T A179968 35,18,7,0,1,8,27,56,70,54,28,8,0,1,8,36,80,125,120,84,32,9,0,1,10,45,
%U A179968 120,210,252,210,120,45,10,0
%N A179968 'AT(n,k)' triangle read by rows. AT(n,k) is the number of aperiodic k-compositions of n.
%C A179968 A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
%C A179968 A k-composition is aperiodic (primitive) if its period is k, or if it is not the concatenation of at least two smaller compositions.
%C A179968 Let AT(n,k) denote the number of aperiodic k-compositions of n.
%C A179968 This sequence is the 'AT(n,k)' triangle read by rows.
%C A179968 If we form the row sum sequence of the 'AT(n, k)' triangle above we get sequence A056278, except for the first term.
%C A179968 If we form the triangle which counts the aperiodic k-compositions of n up to cyclic equivalence, ATE(n, k), and place an extra 0 at the end of each row of the 'ATE(n, k)' triangle, we get sequence A051168 (Lyndon words).
%D A179968 John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [Apparently unpublished as of Mar 27 2017]
%H A179968 Andrew Howroyd, <a href="/A179968/b179968.txt">Table of n, a(n) for n = 1..1275</a>
%H A179968 P. Flajolet and M. Soria, <a href="/A008965/a008965.pdf">The Cycle Construction</a>, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
%F A179968 T(n,k) = k * A051168(n,k). - _Andrew Howroyd_, Mar 26 2017
%F A179968 T(n,k) = (k/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d,k/d). - _Andrew Howroyd_, Mar 26 2017
%F A179968 G.f.: Sum_{k>=1}mu(k)y^k A(x^k)/(1 - y^k A(x^k)) where mu(k) is the Moebius Mu function and A(x) = x/(1-x). - _Geoffrey Critzer_, Aug 05 2022
%e A179968 The triangle begins
%e A179968 1
%e A179968 1, 0
%e A179968 1, 2, 0
%e A179968 1, 2, 3, 0
%e A179968 1, 4, 6, 4, 0
%e A179968 1, 4, 9, 8, 5, 0
%e A179968 1, 6, 15, 20, 15, 6, 0
%e A179968 1, 6, 21, 32, 35, 18, 7, 0
%e A179968 1, 8, 27, 56, 70, 54, 28, 8, 0
%e A179968 1, 8, 36, 80, 125, 120, 84, 32, 9, 0
%e A179968 For example, row 8 is 1, 6, 21, 32, 35, 18, 7, 0.
%e A179968 We have AT(8,2)=6 because there are 6 aperiodic 2-compositions of 8, namely: 17, 71, 26, 62, 35, 53. The remaining 2-composition of 8 is 44, it is not aperiodic.
%e A179968 We have AT(8,3)=21 because all 21 3-compositions of 8 are aperiodic.
%t A179968 T[n_, k_]:=(k/n) * Sum[MoebiusMu[d] Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]; Flatten[Table[T[n, k], {n, 11}, {k, n}]] (* _Indranil Ghosh_, Mar 27 2017, after the formula by _Andrew Howroyd_ *)
%Y A179968 Cf. A051168, A056278.
%K A179968 nonn,tabl
%O A179968 1,5
%A A179968 _John P. McSorley_, Aug 03 2010
%E A179968 a(56)-a(66) from _Andrew Howroyd_, Mar 26 2017