Formulas for A271703

Recurrence

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

Generating function of triangle row polynomials

$L_{n}(x)\ =\ \sum _{0\leq k\leq n}T_{n,k}\,x^{k}$

$L_{n}(x)\ =\ n!\,[t^{n}]\,\exp \left({\frac {tx}{1-t}}\right)$

Hypergeometric series ₁F₁.

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

Kummer U.

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

L⁽¹⁾_n generalized Laguerre polynomials.

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

Reflected Lah polynomials, Hypergeometric series ₂F₀.

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

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

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

$T_{n+k,\,k}\ =\ S_{n}(k)/n!$

Generating rational function of rectangle rows, A253284.

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

$T_{n+k,\,k}\ =\ [x^{k}]\,G_{n}(x)$

Generating exponential function of rectangle columns, A253283.

$R_{n}(x)\ =\ {\frac {d^{n}}{dx^{n}}}\left({\frac {1}{n!}}\left({\frac {x}{1-x}}\right)^{n}\right)$

$T_{n+k,\,n}\ =\ k!\,[x^{k}]R_{n}(x)\quad (n\geq 1)$

Exponential Riordan array.

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

Stirling cycle number, Stirling subset number.

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

Triangle for 0 ≤ k ≤ n:

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

A103371:

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

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

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

Triangle, T_0,0 = 1 and for 0 ≤ k ≤ n and n > 0:

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

Rows of rectangle

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

Columns of rectangle

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

Diagonal of rectangle, C_n Catalan number.

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

Diagonal of rectangle, Γ function.

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

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

Cite this page as

M. F. Hasler and P. Luschny, Formulas for A271703.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Formulas_for_A271703]