A369116 Expansion of (1 - x)^2 * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

1, -1, 1, 0, 1, 1, 3, 4, 9, 15, 29, 53, 100, 186, 352, 663, 1257, 2387, 4547, 8678, 16602, 31818, 61092, 117486, 226277, 436403, 842731, 1629297, 3153466, 6109704, 11848634, 22998892, 44680016, 86869392, 169024094, 329110519, 641254825, 1250261783, 2439155631

Offset: 0

Author

Peter Luschny, Jan 21 2024

Keywords

• sign

Comments

Considering more generally the family of generating functions (1 - x)^n * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)) one finds several sequences related to compositions as indicated in the cross-references.

Crossrefs

Cf. A369115 (n=-2), A186537 left shifted (n=-1), A079500 (n=0), A368279 (n=1), this sequence (n=2).

Programs

Maple

gf := (1 - x)^2 * add(x^j / (1 - add(x^k, k = 1..j)), j = 0..42):
ser := series(gf, x, 40): seq(coeff(ser, x, k), k = 0..38);

Formula

a(n) = A368279(n) - A368279(n-1) where A368279(-1) = 0.