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%I #25 Apr 22 2019 01:50:28
%S 1,-1,2,1,-2,8,-1,2,-8,16,1,-2,8,-16,128,-1,2,-8,16,-128,256,1,-2,8,
%T -16,128,-256,1024,-1,2,-8,16,-128,256,-1024,2048,1,-2,8,-16,128,-256,
%U 1024,-2048,32768,-1,2,-8,16,-128,256,-1024,2048,-32768,65536,1,-2,8,-16,128,-256,1024,-2048,32768
%N 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).
%C 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.
%C The binomial transform of 1, -1/3, 1/5, -1/7, 1/9 is given by the sequence A046161(n)/A001803(n).
%C This sequence of fractions and its higher order differences in the subsequent rows start as:
%C 1, 2/3, 8/15, 16/35, 128/315, 256/693, 1024/3003, 2048/6435, ...
%C -1/3, -2/15, -8/105, -16/315, -128/3465, -256/9009, -1024/45045, ...
%C 1/5, 2/35, 8/315, 16/1155, 128/15015, 256/45045, 1024/255255, ...
%C -1/7, -2/64,-8/693, -16/3003, -128/45045, ...
%C The numerators of this array, read upwards along antidiagonals, define the current sequence.
%F T(n,k) = (-1)^(n-k)*denom(binomial(-1/2,k)). _Peter Luschny_, Nov 21 2012
%e Triangle begins:
%e 1;
%e -1, 2;
%e 1, -2, 8;
%e -1, 2, -8, 16;
%e 1, -2, 8, -16, 128;
%e -1, 2, -8, 16, -128, 256;
%e 1, -2, 8, -16, 128, -256, 1024;
%p A173755 := proc(n,k)
%p local L,i;
%p L := [seq((-1)^i/(2*i+1),i=0..n+k)] ;
%p L := BINOMIAL(L);
%p for i from 1 to n do
%p L := DIFF(L) ;
%p end do:
%p op(1+k,L) ;
%p numer(%) ;
%p end proc: # _R. J. Mathar_, Sep 22 2011
%p A173755 := proc(n, k) local w; w := proc(n) option remember;
%p `if`(n=0,1,2^(padic[ordp](2*n,2))*w(n-1)) end: (-1)^(n-k)*w(k) end:
%p for n from 0 to 8 do seq(A173755(n,k),k=0..n) od; # _Peter Luschny_, Nov 16 2012
%t Table[(-1)^(n - k)*2^(2 k - DigitCount[k, 2, 1]), {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 21 2019 *)
%o (Sage)
%o def A173755(n,k):
%o A005187 = lambda n: A005187(n//2) + n if n > 0 else 0
%o return (-1)^(n-k)*2^A005187(k)
%o for n in (0..8):
%o [A173755(n,k) for k in (0..n)] # _Peter Luschny_, Nov 16 2012
%Y Cf. A046161.
%K tabl,sign
%O 0,3
%A _Paul Curtz_, Feb 23 2010
%E Simpler definition by _Peter Luschny_, Nov 21 2012
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