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.
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.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n,k) = (-1)^(n-k)*denom(binomial(-1/2,k)). Peter Luschny, Nov 21 2012
EXAMPLE
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;
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 *)
PROG
(Sage)
def A173755(n, k):
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
(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
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Paul Curtz, Feb 23 2010
EXTENSIONS
Simpler definition by Peter Luschny, Nov 21 2012
STATUS
approved