A304907 Expansion of x * (d/dx) 1/(1 - Sum_{k>=1} x^k/(1 + x^k)).

0, 1, 2, 9, 16, 35, 84, 161, 312, 639, 1240, 2354, 4536, 8593, 16128, 30360, 56672, 105213, 195174, 360582, 664040, 1220730, 2238324, 4095035, 7479552, 13636750, 24821108, 45114813, 81887008, 148438211, 268763160, 486082263, 878200416, 1585098372, 2858378368, 5149986275

Offset: 0

Author

Ilya Gutkovskiy, May 20 2018

Keywords

• nonn

Comments

Sum of all parts of all Carlitz compositions (compositions without adjacent equal parts) of n.

Links

• Index entries for sequences related to compositions

Crossrefs

Cf. A001787, A003242, A048272, A066186, A066189.

Programs

Mathematica

nmax = 35; CoefficientList[Series[x D[1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), x], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[-(-1)^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[n a[n], {n, 0, 35}]

Formula

a(n) = n*A003242(n).