Some basic sequences, generating functions and closed forms

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Some simple sequences and their generating functions

    \begin{align*} &\mathsf{sequence} &&\mathsf{gen.\; function} &&\mathsf{closed\; form}\\ &\seqd{1,0,0,0,0,0} &&\sumzn{[n=0]} &&1\\ &\seqd{0,\dots,0,1,0,0} &&\sumzn{[n=m]} &&z^m \\ &\seqd{1,1,1,1,1,1} &&\sumzn{} && \fr{1-z}\\ &\seqd{1,-1,1,-1,1,-1} &&\sumzn{(-1)^n} && \fr{1+z}\\ &\seqd{1,0,1,0,1,0} &&\sumzn{[2\backslash n]} && \fr{1-z^2}\\\ &\seqd{1,0,\dots,0,1,0,\dots,0,1,0} &&\sumzn{[m\backslash n]} && \fr{1-z^m}\\ &\seqd{1,2,3,4,5,6} &&\sumzn{(n+1)} && \fr{(1-z)^2}\\ &\text{Triangular numbers}&& && \\ &\seqd{1,3,6,10,15,21} &&\sumzn{\binom{n+2}{2}} && \fr{(1-z)^3}\\ &\text{Tetrahedral numbers}&& && \\ &\seqd{1,4,10,20,35} &&\sumzn{\binom{n+3}{3}} && \fr{(1-z)^4}\\ &\seqd{1,5,15,35,70} &&\sumzn{\binom{n+4}{4}} && \fr{(1-z)^5}\\ &\textstyle\seqd{1,c,\binom{c+1}{2}, \binom{c+2}{3}} &&\displaystyle\sumzn{\binom{c+n-1}{n}} && \fr{(1-z)^c}\\ &\text{equivalent to} &&\sumzn{\binom{n+k}{k}} && \fr{(1-z)^{k+1}}\\ \end{align*}

    \begin{align*} &\seqd{0,1,4,9,16} &&\sumzn{n^2} && \frac{z(z+1)}{(1-z)^3}\\ &\seqd{1,2,4,8,16,32} &&\sumzn{2^n} && \fr{1-2z}\\ &\seqd{1,4,6,4,1,0,0} &&\sumzn{ \binom{4}{n}} && (1+z)^4\\ &\textstyle\seqd{1,c,\binom{c}{2}, \binom{c}{3}} &&\sumzn{\binom{c}{n}} && (1+z)^c\\ &\seqd{1,c,c^2,c^3} &&\sumzn{c^n}&& \fr{1-cz}\\ &\seqd{0,1,\fr{2},\fr{3},\fr{4}} &&\snge[1] \fr{n}z^n && \ln\fr{1-z}\\ &\langle 0,1,-\fr{2},\fr{3},-\fr{4},\dots\rangle &&\snge[1] \frac{-1^{n+1}}{n}z^n && \ln (1+z) \\ &\langle 1,1,\fr{2},\fr{6},\fr{24},\fr{120},\dots\rangle &&\sumzn{\frac{1}{n!}} && e^z\\ &\seqd{0,1,0,\fr{3!},0,\fr{5!}} && \sumzn{\fr{n!}[2\backslash (n+1)]} && \sin{z}\\ &\seqd{1,0,\fr{2!},0,\fr{4!},0} && \sumzn{\fr{n!}[2\backslash (n+2)]} && \cos{z}\\ &\seqd{} && \sumzn{} && \\ \end{align*}

    \begin{align*} &\text{Catalan numbers }C_n&& && \\ &\langle 1,1,2,5,14,42,\dots\rangle && \sum_n \frac{1}{n+1}\binom{2n}{n}z^n&& \frac{1}{2z}(1-\sqrt{1-4z})\\ \end{align*}

    \begin{align*} &\text{sequence} && \text{gen. function} && \text{closed form}\\ &\seqd{1,4,12,32,80,320} && \sumzn{2^n(n+1)} && \fr{(1-2z)^2}\\ &\seqd{1,2c,3c^2,4c^3,5c^4} && \sumzn{(n+1) c^n} && \fr{(1-cz)^2}\\ &\seqd{\overbrace{0,0,\dotsc,0,0}^\text{m zeros},1,1,1} && z^m \sumzn{} && \frac{z^m}{1-z}\\ &\seqd{1,3,8,20,48,112} && && \frac{1-z}{(1-2z)^2}\\ &\seqd{1,2,5,13,33,81} && && \frac{}{}\\ &\seqd{1,2,5,14,41,122,365} && \sumzn{\frac{3^{n-1}+1}{2}} && \frac{z(2-3z)}{(1-3z)(1-z)}\\ &\text{Bell (exponential) numbers}&& && \\ &\seqd{1,2,5,15,52,203} && && \frac{}{}\\ &\seqd{1,3,10,37,151} && && \frac{}{}\\ \end{align*}

All the ways to write an integer n as a sum of 1s, 2s and 3s. Each representation n=a\cdot 1+b\cdot 2+c\cdot 3 is encoded by a monomial x_1^ax_2^bx_3^c that appears in the coefficient polynomial of y^n:

    \begin{align*} &\fr{(1-x_1y)(1-x_2y^2)(1-x_3y^3)}=1+x_1y+(x_1^2+x_2)y^2+(x_1^3+x_1x_2+x_3)y^3 \\ &+(x_1^4+x_1^2x_2+x_2^2+x_1x_3)y^4+(x_1^5+x_1^3x_2+x_1x_2^2+x_1^2x_3+x_2x_3)y^5 \\ &+(x_1^6+x_1^4x_2+x_1^2x_2^2+x_2^3+x_1^3x_3+x_1x_2x_3+x_3^2)y^6+\dots \end{align*}

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