A000750 Expansion of bracket function.

1, -5, 15, -35, 70, -125, 200, -275, 275, 0, -1000, 3625, -9500, 21250, -42500, 76875, -124375, 171875, -171875, 0, 621875, -2250000, 5890625, -13171875, 26343750, -47656250, 77109375, -106562500, 106562500, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

It appears that the (unsigned) sequence is identical to its 5th-order absolute difference. - John W. Layman, Sep 23 2003

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
• Index entries for linear recurrences with constant coefficients, signature (-5, -10, -10, -5).

Crossrefs

Column 5 of A307047.

Cf. A000748, A000749, A001659, A006090, A049016.

Programs

Mathematica

LinearRecurrence[{-5, -10, -10, -5}, {1, -5, 15, -35}, 30] (* Jean-François Alcover, Feb 11 2016 *)

PARI

Vec(1/((1+x)^5-x^5) + O(x^40)) \\ Michel Marcus, Feb 11 2016

PARI

{a(n) = (-1)^n*sum(k=0, n\5, (-1)^k*binomial(n+4, 5*k+4))} \\ Seiichi Manyama, Mar 21 2019

Formula

G.f.: 1/((1+x)^5-x^5).

a(n) = (-1)^n * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+4,5*k+4). - Seiichi Manyama, Mar 21 2019