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A173755 Table read by rows, T(n,k) = (-1)^(n-k)*2^(2*k-bw(k)), where bw(k) is the binary weight of k (A000120). 2
1, -1, 2, 1, -2, 8, -1, 2, -8, 16, 1, -2, 8, -16, 128, -1, 2, -8, 16, -128, 256, 1, -2, 8, -16, 128, -256, 1024, -1, 2, -8, 16, -128, 256, -1024, 2048, 1, -2, 8, -16, 128, -256, 1024, -2048, 32768, -1, 2, -8, 16, -128, 256, -1024, 2048, -32768, 65536, 1, -2, 8, -16, 128, -256, 1024, -2048, 32768 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Old name was: Table of the numerators of the higher order differences of the binomial transform of the Madhava-Gregory-Leibniz series for Pi/4.

The binomial transform of 1, -1/3, 1/5, -1/7, 1/9 is given by the sequence A046161(n)/A001803(n).

This sequence of fractions and its higher order differences in the subsequent rows start as:

1, 2/3, 8/15, 16/35, 128/315, 256/693, 1024/3003, 2048/6435,..

-1/3, -2/15, -8/105, -16/315, -128/3465, -256/9009, -1024/45045,...

1/5, 2/35, 8/315, 16/1155, 128/15015, 256/45045, 1024/255255,...

-1/7, -2/64,-8/693, -16/3003, -128/45045,..

The numerators of this array, read upwards along antidiagonals, define the current sequence.

LINKS

Table of n, a(n) for n=0..63.

FORMULA

T(n,k) = (-1)^(n-k)*denom(binomial(-1/2,k)). Peter Luschny, Nov 21 2012

EXAMPLE

1

-1   2

1  -2   8

-1   2  -8   16

1  -2   8  -16   128

-1   2  -8   16  -128   256

1  -2   8  -16   128  -256  1024

MAPLE

A173755 := proc(n, k)

        local L, i;

        L := [seq((-1)^i/(2*i+1), i=0..n+k)] ;

        L := BINOMIAL(L);

        for i from 1 to n do

                L := DIFF(L) ;

        end do:

        op(1+k, L) ;

        numer(%) ;

end proc: # R. J. Mathar, Sep 22 2011

A173755 := proc(n, k) local w; w := proc(n) option remember;

`if`(n=0, 1, 2^(padic[ordp](2*n, 2))*w(n-1)) end: (-1)^(n-k)*w(k) end:

for n from 0 to 8 do seq(A173755(n, k), k=0..n) od; # Peter Luschny, Nov 16 2012

PROG

(Sage)

def A173755(n, k):

    A005187 = lambda n: A005187(n//2) + n if n > 0 else 0

    return (-1)^(n-k)*2^A005187(k)

for n in (0..8):

    [A173755(n, k) for k in (0..n)]  # Peter Luschny, Nov 16 2012

CROSSREFS

Cf. A046161.

Sequence in context: A030651 A179946 A198757 * A140894 A208747 A221878

Adjacent sequences:  A173752 A173753 A173754 * A173756 A173757 A173758

KEYWORD

tabl,sign

AUTHOR

Paul Curtz, Feb 23 2010

EXTENSIONS

Simpler definition by Peter Luschny, Nov 21 2012

STATUS

approved

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Last modified February 22 15:12 EST 2018. Contains 299454 sequences. (Running on oeis4.)