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# Formulas for A271703

#### Recurrence

${\displaystyle {\begin{cases}T_{n,\,n}=1,\ T_{n,\,k}=0\ (k<0),\\T_{n,\,k}\ =\ T_{n-1,\,k-1}+(n+k-1)T_{n-1,\,k}\end{cases}}}$

#### Generating function of triangle row polynomials

${\displaystyle L_{n}(x)\ =\ \sum _{0\leq k\leq n}T_{n,k}\,x^{k}}$
${\displaystyle L_{n}(x)\ =\ n!\,[t^{n}]\,\exp \left({\frac {tx}{1-t}}\right)}$

#### Hypergeometric series 1F1.

${\displaystyle L_{n}(x)\ =\ n!\,x\,F\left({-n+1 \atop 2}\,{\bigg |}\,-x\right)\quad (n\geq 1)}$

#### Kummer U.

${\displaystyle L_{n}(x)\ =\ (-1)^{n-1}\,x\,U_{1-n,2}(-x)}$

#### L(1)n generalized Laguerre polynomials.

${\displaystyle L_{n}(x)\ =\ (n-1)!\,x\,L_{n-1}^{(1)}(-x)\quad (n\geq 1)}$

#### Reflected Lah polynomials, Hypergeometric series 2F0.

${\displaystyle x^{n}\,L_{n}(1/x)\ =\ F\left({-n+1,\,-n \atop -}\,{\bigg |}\,x\right)}$

#### Generating polynomial of rectangle rows, Stirling cycle number, A254881.

${\displaystyle S_{n}(x)\ =\ \delta _{n,0}+\sum _{k=0}^{2n}x^{k}\sum _{j=0}^{k-1}\left[{n+1 \atop j+1}\right]\,\left[{n \atop k-j}\right]}$
${\displaystyle T_{n+k,\,k}\ =\ S_{n}(k)/n!}$

#### Generating rational function of rectangle rows, A253284.

${\displaystyle G_{n}(x)\ =\ -\sum _{k=0}^{n}{\frac {(n+1)!}{k+1}}{\binom {n+k}{n}}{\binom {n}{k}}(x-1)^{-(n+k+1)}}$
${\displaystyle T_{n+k,\,k}\ =\ [x^{k}]\,G_{n}(x)}$

#### Generating exponential function of rectangle columns, A253283.

${\displaystyle R_{n}(x)\ =\ {\frac {d^{n}}{dx^{n}}}\left({\frac {1}{n!}}\left({\frac {x}{1-x}}\right)^{n}\right)}$
${\displaystyle T_{n+k,\,n}\ =\ k!\,[x^{k}]R_{n}(x)\quad (n\geq 1)}$

#### Exponential Riordan array.

${\displaystyle T_{n,\,k}\ =\ {\frac {n!}{k!}}\left[1\mid {\frac {x}{1-x}}\right]_{n,\ k}}$

#### Stirling cycle number, Stirling subset number.

${\displaystyle T_{n,\,k}\ =\ \sum \limits _{j\,=\,k}^{n}\left[{n \atop j}\right]\left\{{j \atop k}\right\}}$

#### Triangle for 0 ≤ k ≤ n:

${\displaystyle T_{n,\,k}\ =\ (n-k)!{\binom {n}{n-k}}{\binom {n-1}{n-k}}}$

#### A103371:

${\displaystyle {\frac {T_{n,\,k}}{(n-k)!}}\ =\ {\binom {n}{n-k}}{\binom {n-1}{n-k}}}$

#### Little Narayana numbers N(n,k), A090181.

${\displaystyle T_{n,\,k}\ =\ (n-k+1)!\,N_{n,\,k}}$

#### Triangle, T0,0 = 1 and for 0 ≤ k ≤ n and n > 0:

${\displaystyle T_{n,\,k}\ =\ {\frac {n\,k}{(n-k)!}}\left((n-1)^{n-k-1}\right)^{2}\quad (n\geq 1)}$   (Exponent to be underlined?)

#### Rows of rectangle

${\displaystyle T_{n+k,\,k}\ =\ {\frac {(n+k)!}{k!}}{\binom {n+k-1}{k-1}}}$

#### Columns of rectangle

${\displaystyle T_{n+k,\,n}\ =\ {\frac {(n+k)!}{n!}}{\binom {n+k-1}{n-1}}\quad (n\geq 1)}$

#### Diagonal of rectangle, Cn Catalan number.

${\displaystyle T_{2n,\,n}\ =\ (n+1)!{\binom {2n-1}{n}}\,C_{n}}$

#### Diagonal of rectangle, Γ function.

${\displaystyle T_{2n,\,n}\ =\ {\frac {2}{n}}{\frac {\Gamma (2n)^{2}}{\Gamma (n)^{3}}}\quad (n\geq 1)}$

### Authorship

This page was created by M. F. Hasler on 28 August 2017, 21:15 (UTC), based on a MathJax page created by Peter Luschny