A001659 Expansion of bracket function. (Formerly M1433 N0567)

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

1,4

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.

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

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 *)

Prog

(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)

Crossrefs

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

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

Sequence in context: A337282 A366117 A027929 * A088921 A005183 A005348

Adjacent sequences: A001656 A001657 A001658 * A001660 A001661 A001662

Keyword

sign

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Jun 14 2003

Status

approved