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A329738 Number of compositions of n whose run-lengths are all equal.

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