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Andrew Howroyd, <a href="/A329738/b329738.txt">Table of n, a(n) for n = 0..1000</a>

(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

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Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329742, A329743, A329745, A329766.

allocated for Gus WisemanNumber of compositions of n whose run-lengths are all equal.

1, 1, 2, 4, 6, 8, 19, 24, 45, 75, 133, 215, 401, 662, 1177, 2035, 3587, 6190, 10933, 18979, 33339, 58157, 101958, 178046, 312088, 545478, 955321, 1670994, 2925717, 5118560, 8960946, 15680074, 27447350, 48033502, 84076143, 147142496, 257546243, 450748484, 788937192

0,3

A composition of n is a finite sequence of positive integers with sum n.

a(n) = Sum_{d|n} A003242(d).

a(n) = A329745(n) + A000005(n).

The a(1) = 1 through a(6) = 19 compositions:

(1) (2) (3) (4) (5) (6)

(11) (12) (13) (14) (15)

(21) (22) (23) (24)

(111) (31) (32) (33)

(121) (41) (42)

(1111) (131) (51)

(212) (123)

(11111) (132)

(141)

(213)

(222)

(231)

(312)

(321)

(1122)

(1212)

(2121)

(2211)

(111111)

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], SameQ@@Length/@Split[#]&]], {n, 0, 10}]

Compositions with relatively prime run-lengths are A000740.

Compositions with equal multiplicities are A098504.

Compositions with equal differences are A175342.

Compositions with distinct run-lengths are A329739.

Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329742, A329743, A329745.

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Gus Wiseman, Nov 20 2019

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