A389349 Expansion of (1/x) * Series_Reversion( x * (1 - x^3 / (1 - x)^3) ).

1, 0, 0, 1, 3, 6, 14, 42, 126, 358, 1026, 3048, 9204, 27846, 84630, 259409, 800907, 2484636, 7739840, 24212061, 76040223, 239629646, 757460958, 2401086726, 7631264270, 24312676830, 77630514390, 248385358330, 796255536546, 2557146018942, 8225904163262, 26502731415474, 85513541680182

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(n-1,n-3*k).

a(n) = (1/(n+1)) * [x^n] 1/(1 - x^3 / (1 - x)^3)^(n+1).

Mathematica

Table[SeriesCoefficient[1/(1-x^3/(1-x)^3)^(n+1), {x, 0, n}]/(n+1), {n, 0, 30}] (* Vincenzo Librandi, Oct 18 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1-x^3/(1-x)^3))/x)
(Magma) [1/(n+1)*&+[Binomial(n+k, k)*Binomial(n-1, n-3*k): k in [0..Floor(n/3)]]: n in [0..35]]; // Vincenzo Librandi, Oct 18 2025

Crossrefs

Cf. A389250.

Sequence in context: A101162 A274054 A059741 * A054099 A257320 A318344

Adjacent sequences: A389346 A389347 A389348 * A389350 A389351 A389352

Keyword

nonn

Author

Seiichi Manyama, Oct 01 2025

Status

approved