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#### 7.19.2 Factorials

Although factorials can be computed from the Gamma function, using the relation n! = \Gamma(n+1) for non-negative integer n, it is usually more efficient to call the functions in this section, particularly for small values of n, whose factorial values are maintained in hardcoded tables.

Function: double gsl_sf_fact (unsigned int n)
Function: int gsl_sf_fact_e (unsigned int n, gsl_sf_result * result)

These routines compute the factorial n!. The factorial is related to the Gamma function by n! = \Gamma(n+1). The maximum value of n such that n! is not considered an overflow is given by the macro GSL_SF_FACT_NMAX and is 170.

Function: double gsl_sf_doublefact (unsigned int n)
Function: int gsl_sf_doublefact_e (unsigned int n, gsl_sf_result * result)

These routines compute the double factorial n!! = n(n-2)(n-4) \dots. The maximum value of n such that n!! is not considered an overflow is given by the macro GSL_SF_DOUBLEFACT_NMAX and is 297.

Function: double gsl_sf_lnfact (unsigned int n)
Function: int gsl_sf_lnfact_e (unsigned int n, gsl_sf_result * result)

These routines compute the logarithm of the factorial of n, \log(n!). The algorithm is faster than computing \ln(\Gamma(n+1)) via gsl_sf_lngamma for n < 170, but defers for larger n.

Function: double gsl_sf_lndoublefact (unsigned int n)
Function: int gsl_sf_lndoublefact_e (unsigned int n, gsl_sf_result * result)

These routines compute the logarithm of the double factorial of n, \log(n!!).

Function: double gsl_sf_choose (unsigned int n, unsigned int m)
Function: int gsl_sf_choose_e (unsigned int n, unsigned int m, gsl_sf_result * result)

These routines compute the combinatorial factor n choose m = n!/(m!(n-m)!)

Function: double gsl_sf_lnchoose (unsigned int n, unsigned int m)
Function: int gsl_sf_lnchoose_e (unsigned int n, unsigned int m, gsl_sf_result * result)

These routines compute the logarithm of n choose m. This is equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).

Function: double gsl_sf_taylorcoeff (int n, double x)
Function: int gsl_sf_taylorcoeff_e (int n, double x, gsl_sf_result * result)

These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.

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