def get_recursion_row(self,i):
r = self.coefficients()
d = self.order()
row = []
#var = self.__polynomialRing.gens()[-1]
### First summation part
#row += [sum([falling_factorial(k,l)*self.base().sequence(r[l],i-k+l) for l in range(0,k+1)]) for k in range(0,min(i,d))]
#if(i<d):
# ## Second summation part
# row += [sum([falling_factorial(k+i,l)*self.base().sequence(r[l], l-k) for l in range(k,i+k+1)]) for k in range(d-i)]
# ## Third summation part
# row += [self.forward(k)(**{str(var):i}) for k in range(d-i,d+1)]
#else:
# ## Second summation part
# row += [self.backward(i-d-k)(**{str(var):i}) for k in range(i-d)]
# ## Third summation part
# row += [self.forward(k)(**{str(var):i}) for k in range(0,d+1)]
## First summation part
row += [sum([falling_factorial(k,l)*self.base().sequence(r[l],i-k+l) for l in range(0,k+1)]) for k in range(0,min(i,d))]
if(i<d):
## Second summation part
row += [sum([falling_factorial(k+i,l)*self.base().sequence(r[l], l-k) for l in range(k,i+k+1)]) for k in range(d-i)]
## Third summation part
row += [self.__eval_pol(self.forward(k),i) for k in range(d-i,d+1)]
else:
## Second summation part
row += [self.__eval_pol(self.backward(i-d-k),i) for k in range(i-d)]
## Third summation part
row += [self.__eval_pol(self.forward(k),i) for k in range(0,d+1)]
return row
def backward(self, n):
if(n < 0):
raise IndexError("Backward polynomials have only positive index")
elif(self.is_zero()):
return 0
var = self.__polynomialRing.gens()[-1]
return sum([self.base().sequence(self.coefficient(l), l+n)*falling_factorial(var-n, l) for l in range(0,self.order()+1)])
def _inverse_egf(n):
if(n == 0):
return 0
elif(n == 1):
return 1/f(1)
else:
result = 0
for k in range(1,n):
poly = bell_polynomial(n-1,k)
extra = ((-1)**k)*falling_factorial(n+k-1, k)
if(k == n-1):
evaluation = poly(x=f(2)/2/f(1))
else:
evaluation = poly(**{"x%d" %(i) : f(i+2)/(i+2)/f(1) for i in range(n-k)})
result += extra*evaluation
return (1/(f(1)**n)) * result
def _sage_(self):
import sage.all as sage
return sage.falling_factorial(self.args[0]._sage_(),
self.args[1]._sage_())
def _sage_(self):
import sage.all as sage
return sage.falling_factorial(self.args[0]._sage_(),
self.args[1]._sage_())
