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).

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, -65536, 262144

Offset: 0

Author

Paul Curtz, Feb 23 2010

Keywords

• tabl
• sign

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, ...

-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.

Examples

			Triangle begins:
   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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A001803, A005187, A046161.

Programs

Magma

A173755:= func< n,k | (-1)^(n-k)*2^(k + Valuation(Factorial(k), 2)) >;
[A173755(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 24 2024

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

Mathematica

Table[(-1)^(n - k)*2^(2 k - DigitCount[k, 2, 1]), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 21 2019 *)

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

Formula

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

Extensions

Simpler definition by Peter Luschny, Nov 21 2012