A006090 Expansion of bracket function.

1, -6, 21, -56, 126, -252, 463, -804, 1365, -2366, 4368, -8736, 18565, -40410, 87381, -184604, 379050, -758100, 1486675, -2884776, 5592405, -10919090, 21572460, -43144920, 87087001, -176565486, 357913941, -723002336

Offset: 0

Author

David Callan

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2, issue 4, (1964), 241-260.
• Problems Drive, Eureka, 37 (1974), 8-11, 32-33, 24-27. (Annotated scanned copy)
• Index entries for linear recurrences with constant coefficients, signature (-6,-15,-20,-15,-6).

Crossrefs

Column 6 of A307047.

Cf. A000748, A000749, A000750, A001659.

Programs

Mathematica

CoefficientList[Series[1/((1+x)^6-x^6),{x,0,30}],x] (* or *) LinearRecurrence[ {-6,-15,-20,-15,-6},{1,-6,21,-56,126},31] (* Harvey P. Dale, Oct 14 2016 *)

PARI

x='x+O('x^50); Vec(1/((1+x)^6-x^6)) \\ G. C. Greubel, Jul 02 2017

Formula

G.f.: 1/((1+x)^6-x^6).

a(n) = (-1)^n * Sum_{k=0..floor(n/6)} binomial(n+5,6*k+5). - Seiichi Manyama, Aug 05 2024