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########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{b} =} bincoeff (@var{n}, @var{k})
## Return the binomial coefficient of @var{n} and @var{k}.
##
## The binomial coefficient is defined as
## @tex
## $$
## {n \choose k} = {n (n-1) (n-2) \cdots (n-k+1) \over k!}
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## / \
## | n | n (n-1) (n-2) @dots{} (n-k+1)
## | | = -------------------------
## | k | k!
## \ /
## @end group
## @end example
##
## @end ifnottex
## For example:
##
## @example
## @group
## bincoeff (5, 2)
## @result{} 10
## @end group
## @end example
##
## In most cases, the @code{nchoosek} function is faster for small
## scalar integer arguments. It also warns about loss of precision for
## big arguments.
##
## @seealso{nchoosek}
## @end deftypefn
function b = bincoeff (n, k)
if (nargin != 2)
print_usage ();
endif
[retval, n, k] = common_size (n, k);
if (retval > 0)
error ("bincoeff: N and K must be of common size or scalars");
endif
if (iscomplex (n) || iscomplex (k))
error ("bincoeff: N and K must not be complex");
endif
b = zeros (size (n));
ok = (k >= 0) & (k == fix (k)) & (! isnan (n));
b(! ok) = NaN;
n_int = (n == fix (n));
idx = n_int & (n < 0) & ok;
b(idx) = (-1) .^ k(idx) .* exp (gammaln (abs (n(idx)) + k(idx))
- gammaln (k(idx) + 1)
- gammaln (abs (n(idx))));
idx = (n >= k) & ok;
b(idx) = exp (gammaln (n(idx) + 1)
- gammaln (k(idx) + 1)
- gammaln (n(idx) - k(idx) + 1));
idx = (! n_int) & (n < k) & ok;
b(idx) = (1/pi) * exp (gammaln (n(idx) + 1)
- gammaln (k(idx) + 1)
+ gammaln (k(idx) - n(idx))
+ log (sin (pi * (n(idx) - k(idx) + 1))));
## Clean up rounding errors.
b(n_int) = round (b(n_int));
idx = ! n_int;
b(idx) = real (b(idx));
endfunction
%!assert (bincoeff (4, 2), 6)
%!assert (bincoeff (2, 4), 0)
%!assert (bincoeff (-4, 2), 10)
%!assert (bincoeff (5, 2), 10)
%!assert (bincoeff (50, 6), 15890700)
%!assert (bincoeff (0.4, 2), -.12, 8*eps)
%!assert (bincoeff ([4 NaN 4], [-1, 2, 2.5]), NaN (1, 3))
## Test input validation
%!error <Invalid call> bincoeff ()
%!error <Invalid call> bincoeff (1)
%!error bincoeff (ones (3), ones (2))
%!error bincoeff (ones (2), ones (3))
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