o 3Ih##@sddlmZmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZmZddlmZddlmZddlZddlmZdd Zd d Zd d ZeddgdedefddZdedefddZ edddddddZ!dS))fzerofrom_int from_rationalfonefhalfbitcountto_intmpf_mulmpf_divmpf_submpf_addmpf_sqrtmpf_pi mpf_cosh_sinhmpf_cosmpf_sin)_sqrt_mod_prime_poweris_quad_residue) deprecated)recurrence_memoNcountcCsd}dg|adg|at|dd}td|D]}t|dkr6t||||D] }t|dkr5|t|<q)qtd|D]4}t|dkrO|t|<|dt|<q._Mr,(rz Input too bigr+2)rZ cache_lengthglobalsr)rr6rAloglog2ramaxceil ValueErrorrr r rrrr r8rTr rabsrr)r9pbitsr;r>rhbigsmallrcerMrYrKrLqrMdr'rdr( _partitions`       rzzpThe `sympy.ntheory.partitions_.npartitions` has been moved to `sympy.functions.combinatorial.numbers.partition`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetFcCsddlm}||S)a  Calculate the partition function P(n), i.e. the number of ways that n can be written as a sum of positive integers. .. deprecated:: 1.13 The ``npartitions`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.partition` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. P(n) is computed using the Hardy-Ramanujan-Rademacher formula [1]_. The correctness of this implementation has been tested through $10^{10}$. Examples ======== >>> from sympy.functions.combinatorial.numbers import partition >>> partition(25) 1958 References ========== .. [1] https://mathworld.wolfram.com/PartitionFunctionP.html r) partition)%sympy.functions.combinatorial.numbersr})r9verbosefunc_partitionr'r'r( npartitionss !r)F)" mpmath.libmprrrrrrrr r r r r rrrrresidue_ntheoryrrsympy.utilities.decoratorrsympy.utilities.memoizationrr6 itertoolsrr)r8rTrrZrzrr'r'r'r(s"H   W M