r"""
Python module for OEIS sequence number A001850.

Central Delannoy numbers: Sum_{k=0..n} C(n,k)*C(n+k,k).

Examples of use.
-----------------------------------------------------------------------
>>> from a001850 import *
>>> print a001850_list(10)
[1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563]
>>> print a001850_offset
0
>>> for x in a001850_list_pairs(6):
...     print x
...
(0, 1)
(1, 3)
(2, 13)
(3, 63)
(4, 321)
(5, 1683)
>>> a001850_list_upto(10000)
[1, 3, 13, 63, 321, 1683, 8989]
>>> print a001850(5)
1683
-----------------------------------------------------------------------
"""

from itertools import islice, izip, takewhile, count

__all__ = ('a001850_offset', 'a001850_list', 'a001850_list_pairs', 'a001850_list_upto', 'a001850', 'a001850_gen')
__author__ = 'Nick Hobson <nickh@qbyte.org>'

a001850_offset = offset = 0

def a001850_gen():
    """Generator function for OEIS sequence A001850."""
    x, y = 1, 3
    for n in count(2):
        yield x
        x, y = y, (3*(2*n-1)*y - (n-1)*x) / n

def a001850_list(n):
    """Returns a list of the first n >= 0 terms."""
    if n < 0: raise ValueError, 'Input must be a non-negative integer'
    return list(islice(a001850_gen(), n))

def a001850_list_pairs(n):
    """Returns a list of tuples (n, a(n)) of the first n >= 0 terms."""
    if n < 0: raise ValueError, 'Input must be a non-negative integer'
    return list(izip(xrange(offset, n+offset), a001850_gen()))

def a001850_list_upto(m):
    """Returns a list of all terms not exceeding m >= 0."""
    if m < 0: raise ValueError, 'Input must be a non-negative integer'
    return list(takewhile(lambda t: t <= m, a001850_gen()))

def a001850(n):
    """Returns the term with index n >= 0; offset 0."""
    if n < offset: raise ValueError, 'Input must be an integer >= offset = ' + str(offset)
    return list(islice(a001850_gen(), n-offset, n-offset+1)).pop()