Zeta and Related Functions - 26.3 Lattice Paths: Binomial Coefficients

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
26.3.E1 ( m n ) = ( m m - n ) binomial π‘š 𝑛 binomial π‘š π‘š 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\genfrac{(}{)}{0.0pt% }{}{m}{m-n}}}
\binom{m}{n} = \binom{m}{m-n}		 m β‰₯ n π‘š 𝑛 {\displaystyle{\displaystyle m\geq n}} 
binomial(m,n) = binomial(m,m - n)

Binomial[m,n] == Binomial[m,m - n]
Failure Successful Successful [Tested: 6] Successful [Tested: 6]
26.3.E1 ( m m - n ) = m ! ( m - n ) ! ⁒ n ! binomial π‘š π‘š 𝑛 π‘š π‘š 𝑛 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{m-n}=\frac{m!}{(m-n)!\,% n!}}}
\binom{m}{m-n} = \frac{m!}{(m-n)!\,n!}		 m β‰₯ n π‘š 𝑛 {\displaystyle{\displaystyle m\geq n}} 
binomial(m,m - n) = (factorial(m))/(factorial(m - n)*factorial(n))

Binomial[m,m - n] == Divide[(m)!,(m - n)!*(n)!]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 6]
26.3.E2 ( m n ) = 0 binomial π‘š 𝑛 0 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=0}}
\binom{m}{n} = 0		 n > m 𝑛 π‘š {\displaystyle{\displaystyle n>m}} 
binomial(m,n) = 0

Binomial[m,n] == 0
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
26.3.E3 βˆ‘ n = 0 m ( m n ) ⁒ x n = ( 1 + x ) m superscript subscript 𝑛 0 π‘š binomial π‘š 𝑛 superscript π‘₯ 𝑛 superscript 1 π‘₯ π‘š {\displaystyle{\displaystyle\sum_{n=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{n}x^{n}=(% 1+x)^{m}}}
\sum_{n=0}^{m}\binom{m}{n}x^{n} = (1+x)^{m}		 

sum(binomial(m,n)*(x)^(n), n = 0..m) = (1 + x)^(m)

Sum[Binomial[m,n]*(x)^(n), {n, 0, m}, GenerateConditions->None] == (1 + x)^(m)
Successful Successful - Successful [Tested: 0]
26.3.E4 βˆ‘ m = 0 ∞ ( m + n m ) ⁒ x m = 1 ( 1 - x ) n + 1 superscript subscript π‘š 0 binomial π‘š 𝑛 π‘š superscript π‘₯ π‘š 1 superscript 1 π‘₯ 𝑛 1 {\displaystyle{\displaystyle\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{m+n}{m}% x^{m}=\frac{1}{(1-x)^{n+1}}}}
\sum_{m=0}^{\infty}\binom{m+n}{m}x^{m} = \frac{1}{(1-x)^{n+1}}		 | x | < 1 π‘₯ 1 {\displaystyle{\displaystyle|x|<1}} 
sum(binomial(m + n,m)*(x)^(m), m = 0..infinity) = (1)/((1 - x)^(n + 1))

Sum[Binomial[m + n,m]*(x)^(m), {m, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - x)^(n + 1)]
Successful Successful - Successful [Tested: 3]
26.3.E5 ( m n ) = ( m - 1 n ) + ( m - 1 n - 1 ) binomial π‘š 𝑛 binomial π‘š 1 𝑛 binomial π‘š 1 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\genfrac{(}{)}{0.0pt% }{}{m-1}{n}+\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}}}
\binom{m}{n} = \binom{m-1}{n}+\binom{m-1}{n-1}		 m β‰₯ n , n β‰₯ 1 formulae-sequence π‘š 𝑛 𝑛 1 {\displaystyle{\displaystyle m\geq n,n\geq 1}} 
binomial(m,n) = binomial(m - 1,n)+binomial(m - 1,n - 1)

Binomial[m,n] == Binomial[m - 1,n]+Binomial[m - 1,n - 1]
Successful Successful - Successful [Tested: 6]
26.3.E6 ( m n ) = m n ⁒ ( m - 1 n - 1 ) binomial π‘š 𝑛 π‘š 𝑛 binomial π‘š 1 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\frac{m}{n}\genfrac{% (}{)}{0.0pt}{}{m-1}{n-1}}}
\binom{m}{n} = \frac{m}{n}\binom{m-1}{n-1}		 m β‰₯ n , n β‰₯ 1 formulae-sequence π‘š 𝑛 𝑛 1 {\displaystyle{\displaystyle m\geq n,n\geq 1}} 
binomial(m,n) = (m)/(n)*binomial(m - 1,n - 1)

Binomial[m,n] == Divide[m,n]*Binomial[m - 1,n - 1]
Successful Successful - Successful [Tested: 6]
26.3.E6 m n ⁒ ( m - 1 n - 1 ) = m - n + 1 n ⁒ ( m n - 1 ) π‘š 𝑛 binomial π‘š 1 𝑛 1 π‘š 𝑛 1 𝑛 binomial π‘š 𝑛 1 {\displaystyle{\displaystyle\frac{m}{n}\genfrac{(}{)}{0.0pt}{}{m-1}{n-1}=\frac% {m-n+1}{n}\genfrac{(}{)}{0.0pt}{}{m}{n-1}}}
\frac{m}{n}\binom{m-1}{n-1} = \frac{m-n+1}{n}\binom{m}{n-1}		 m β‰₯ n , n β‰₯ 1 formulae-sequence π‘š 𝑛 𝑛 1 {\displaystyle{\displaystyle m\geq n,n\geq 1}} 
(m)/(n)*binomial(m - 1,n - 1) = (m - n + 1)/(n)*binomial(m,n - 1)

Divide[m,n]*Binomial[m - 1,n - 1] == Divide[m - n + 1,n]*Binomial[m,n - 1]
Successful Successful - Successful [Tested: 6]
26.3.E7 ( m + 1 n + 1 ) = βˆ‘ k = n m ( k n ) binomial π‘š 1 𝑛 1 superscript subscript π‘˜ 𝑛 π‘š binomial π‘˜ 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m+1}{n+1}=\sum_{k=n}^{m}% \genfrac{(}{)}{0.0pt}{}{k}{n}}}
\binom{m+1}{n+1} = \sum_{k=n}^{m}\binom{k}{n}		 m β‰₯ n , n β‰₯ 0 formulae-sequence π‘š 𝑛 𝑛 0 {\displaystyle{\displaystyle m\geq n,n\geq 0}} 
binomial(m + 1,n + 1) = sum(binomial(k,n), k = n..m)

Binomial[m + 1,n + 1] == Sum[Binomial[k,n], {k, n, m}, GenerateConditions->None]
Successful Successful - Successful [Tested: 6]
26.3.E8 ( m n ) = βˆ‘ k = 0 n ( m - n - 1 + k k ) binomial π‘š 𝑛 superscript subscript π‘˜ 0 𝑛 binomial π‘š 𝑛 1 π‘˜ π‘˜ {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}% \genfrac{(}{)}{0.0pt}{}{m-n-1+k}{k}}}
\binom{m}{n} = \sum_{k=0}^{n}\binom{m-n-1+k}{k}		 m β‰₯ n , n β‰₯ 0 formulae-sequence π‘š 𝑛 𝑛 0 {\displaystyle{\displaystyle m\geq n,n\geq 0}} 
binomial(m,n) = sum(binomial(m - n - 1 + k,k), k = 0..n)

Binomial[m,n] == Sum[Binomial[m - n - 1 + k,k], {k, 0, n}, GenerateConditions->None]
Successful Successful -
Failed [3 / 6]
Result: Indeterminate
Test Values: {Rule[m, 1], Rule[n, 1]}


Result: Indeterminate
Test Values: {Rule[m, 2], Rule[n, 2]}

... skip entries to safe data
26.3.E9 ( n 0 ) = ( n n ) binomial 𝑛 0 binomial 𝑛 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{0}=\genfrac{(}{)}{0.0pt% }{}{n}{n}}}
\binom{n}{0} = \binom{n}{n}		 

binomial(n,0) = binomial(n,n)

Binomial[n,0] == Binomial[n,n]
Successful Successful - Successful [Tested: 3]
26.3.E9 ( n n ) = 1 binomial 𝑛 𝑛 1 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{n}=1}}
\binom{n}{n} = 1		 

binomial(n,n) = 1

Binomial[n,n] == 1
Successful Successful - Successful [Tested: 3]
26.3.E10 ( m n ) = βˆ‘ k = 0 n ( - 1 ) n - k ⁒ ( m + 1 k ) binomial π‘š 𝑛 superscript subscript π‘˜ 0 𝑛 superscript 1 𝑛 π‘˜ binomial π‘š 1 π‘˜ {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{m}{n}=\sum_{k=0}^{n}(-1)^{% n-k}\genfrac{(}{)}{0.0pt}{}{m+1}{k}}}
\binom{m}{n} = \sum_{k=0}^{n}(-1)^{n-k}\binom{m+1}{k}		 m β‰₯ n , n β‰₯ 0 formulae-sequence π‘š 𝑛 𝑛 0 {\displaystyle{\displaystyle m\geq n,n\geq 0}} 
binomial(m,n) = sum((- 1)^(n - k)*binomial(m + 1,k), k = 0..n)

Binomial[m,n] == Sum[(- 1)^(n - k)*Binomial[m + 1,k], {k, 0, n}, GenerateConditions->None]
Successful Failure - Successful [Tested: 6]
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