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%I #9 Dec 30 2020 19:44:22
%S 1,1,2,4,6,8,19,24,45,75,133,215,401,662,1177,2035,3587,6190,10933,
%T 18979,33339,58157,101958,178046,312088,545478,955321,1670994,2925717,
%U 5118560,8960946,15680074,27447350,48033502,84076143,147142496,257546243,450748484,788937192
%N Number of compositions of n whose run-lengths are all equal.
%C A composition of n is a finite sequence of positive integers with sum n.
%H Andrew Howroyd, <a href="/A329738/b329738.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{d|n} A003242(d).
%F a(n) = A329745(n) + A000005(n).
%e The a(1) = 1 through a(6) = 19 compositions:
%e (1) (2) (3) (4) (5) (6)
%e (11) (12) (13) (14) (15)
%e (21) (22) (23) (24)
%e (111) (31) (32) (33)
%e (121) (41) (42)
%e (1111) (131) (51)
%e (212) (123)
%e (11111) (132)
%e (141)
%e (213)
%e (222)
%e (231)
%e (312)
%e (321)
%e (1122)
%e (1212)
%e (2121)
%e (2211)
%e (111111)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@Split[#]&]],{n,0,10}]
%o (PARI) seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); concat([1], vector(n, k, sumdiv(k, d, b[d])))} \\ _Andrew Howroyd_, Dec 30 2020
%Y Compositions with relatively prime run-lengths are A000740.
%Y Compositions with equal multiplicities are A098504.
%Y Compositions with equal differences are A175342.
%Y Compositions with distinct run-lengths are A329739.
%Y Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329745, A329766.
%K nonn
%O 0,3
%A _Gus Wiseman_, Nov 20 2019