Combinatorial Analysis - 26.11 Integer Partitions: Compositions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
26.11.E2	${\displaystyle{\displaystyle c_{m}\left(0\right)=\delta_{0,m}}}$ \ncompositions[m]@{0} = \Kroneckerdelta{0}{m}		numbcomp(0, m) = KroneckerDelta[0, m]	Error	Error	Missing Macro Error	-	-
26.11.E3	${\displaystyle{\displaystyle c_{m}\left(n\right)=\genfrac{(}{)}{0.0pt}{}{n-1}{m-1}}}$ \ncompositions[m]@{n} = \binom{n-1}{m-1}		numbcomp(n, m) = binomial(n - 1,m - 1)	Error	Error	Missing Macro Error	-	-
26.11.E4	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}c_{m}\left(n\right)q^{n}=\frac{q^{m}}{(1-q)^{m}}}}$ \sum_{n=0}^{\infty}\ncompositions[m]@{n}q^{n} = \frac{q^{m}}{(1-q)^{m}}		sum(numbcomp(n, m)*(q)^(n), n = 0..infinity) = ((q)^(m))/((1 - q)^(m))	Error	Error	Missing Macro Error	-	-
26.11#Ex1	${\displaystyle{\displaystyle F_{0}=0}}$ F_{0} = 0		F[0] = 0	Subscript[F, 0] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
26.11#Ex2	${\displaystyle{\displaystyle F_{1}=1}}$ F_{1} = 1		F[1] = 1	Subscript[F, 1] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
26.11#Ex3	${\displaystyle{\displaystyle F_{n}=F_{n-1}+F_{n-2}}}$ F_{n} = F_{n-1}+F_{n-2}	${\displaystyle{\displaystyle n\geq 2}}$	F[n] = F[n - 1]+ F[n - 2]	Subscript[F, n] == Subscript[F, n - 1]+ Subscript[F, n - 2]	Skipped - no semantic math	Skipped - no semantic math	-	-
26.11.E7	${\displaystyle{\displaystyle F_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\,\sqrt{5}}}}$ F_{n} = \frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\,\sqrt{5}}		F[n] = ((1 +sqrt(5))^(n)-(1 -sqrt(5))^(n))/((2)^(n)*sqrt(5))	Subscript[F, n] == Divide[(1 +Sqrt[5])^(n)-(1 -Sqrt[5])^(n),(2)^(n)*Sqrt[5]]	Skipped - no semantic math	Skipped - no semantic math	-	-