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A001659
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Expansion of bracket function.
(Formerly M1433 N0567)
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Inverse binomial transform of A006218.
The g.f. -(-1+5*z-8*z**2+z**3+5*z**4+z**5)/(1-6*z+13*z**2-10*z**3-z**4+z**5) conjectured by S. Plouffe in his 1992 dissertation is wrong.
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REFERENCES
| H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2 (1964), 241-260. Math. Rev. 30 #1090
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).
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| 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 (vladeta(AT)eunet.rs), 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
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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)
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CROSSREFS
| Cf. A000748, A000749, A000750, A006090, A006218.
Equals A038200(n-1) + A038200(n), n>1.
Sequence in context: A110320 A108890 A027929 * A088921 A005183 A005348
Adjacent sequences: A001656 A001657 A001658 * A001660 A001661 A001662
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Michael Somos, Jun 14 2003
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