I'm looking to see if there is a standard notation for the following other than Binomial coefficient notation...
The triangular numbers $1,3,6,10,15,21,...$ are notated as $T_n$ and represented as $\binom{n+1}{2}$
The tetrahedral numbers $1,4,10,20,35,56,...$ are notated (per Wikipedia) as $Te_n=\sum{T_n}$ and represented binomially as $\binom{n+2}{3}$
Then there are the 5-cell numbers $1,5,15,35,70,126,...$. These would naturally be represented as $\sum{Te_n}$, but was wondering if there was any other representation of this family of sequences of higher order n-simplex numbers...
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$\begingroup$As you already know, these simplicial polytopic numbers are readily expressed as binomial coefficients. The $n^{\text{th}}$ number in the sequence of $d$-dimensional simplex figurates is $\binom{n+d-1}{d}$, which is also called a stars-and-bars number as it also arises in that context.
Some authors employ an emphasized variant of the binomial coefficient notation to mean this closely related expression. For example the OeisWiki page linked above uses doubled parentheses:
$$ \left(\! \! \binom{n}{d} \! \! \right) := \binom{n+d-1}{d} $$
while the WolframMathworld page uses single bold parentheses for what it terms "multichoose". When I try to recreate these bolded delimiters with $\LaTeX$ here, the effect is rather too subtle:
$$ \mathbf{\bigg{(}} \genfrac {}{}{0pt}{}{n}{d} \mathbf{\bigg{)}} $$
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