A001659 Expansion of bracket function.

1, 1, -1, 2, -5, 13, -33, 80, -184, 402, -840, 1699, -3382, 6750, -13716, 28550, -60587, 129579, -275915, 579828, -1197649, 2431775, -4870105, 9672634, -19173013, 38151533, -76521331, 154941608, -316399235, 649807589, -1337598675, 2751021907, -5640238583, 11513062785, -23389948481, 47310801199, -95345789479, 191616365385

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

Comments

Inverse binomial transform of A006218.

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2, issue 4, (1964), 241-260.

Crossrefs

Cf. A000005, A000748, A000749, A000750, A006090, A006218.

Equals A038200(n-1) + A038200(n), n>1.

Programs

Mathematica

Table[Sum[(-1)^(n - k)*Binomial[n, k]*Sum[Floor[k/j], {j, 1, k}], {k, 0, n}], {n, 1, 50}] (* G. C. Greubel, Jul 02 2017 *)

PARI

a(n)=sum(j=0,n,(-1)^(n-j)*binomial(n,j)*sum(k=1,j,j\k))

PARI

a(n)=polcoeff(sum(k=1,n,x^k/((1+x)^k-x^k),x*O(x^n)),n)

Formula

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

G.f.: Sum_{k>0} x^k/((1+x)^k-x^k).

G.f.: Sum_{k>0} tau(k)*x^k/(1+x)^k. - Vladeta Jovovic, Jun 24 2003

G.f.: Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1+x). - Joerg Arndt, Jan 30 2011

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*tau(k). - Ridouane Oudra, Aug 21 2021

Extensions

Edited by Michael Somos, Jun 14 2003