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# User:Jaume Oliver Lafont/C((d+1)n+d,d)

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}={\frac {2\pi }{3{\sqrt {3}}}}={\frac {4}{9}}\sin({\frac {\pi }{3}})\pi ={\frac {\frac {\pi }{3}}{sin({\frac {\pi }{3}})}}={\frac {1}{sinc({\frac {\pi }{3}})}}}$ = 1/A086089 [1] Centered nonagonal numbers A060544

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {4n+3}{3}}}={\frac {3}{4}}log(4)={\frac {3}{4}}(2log(2))={\frac {3}{4}}log(2^{2})}$ [2] Odd tetrahedral numbers A015219

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {5n+4}{4}}}={\frac {4}{5}}{\sqrt {{\frac {2}{5}}(25-11{\sqrt {5}})}}\pi ={\frac {16}{25}}(3sin({\frac {2\pi }{5}})-4sin({\frac {\pi }{5}}))\pi }$ [3] [4] [5] A151989

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {6n+5}{5}}}={\frac {5}{6}}(16log(2)-9log(3))={\frac {5}{6}}log({\frac {2^{16}}{3^{9}}})}$ [6]

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {7n+6}{6}}}={\frac {6}{7}}(10\tan({\frac {\pi }{14}})-5\tan({\frac {3\pi }{14}})+\cot({\frac {\pi }{7}}))\pi ={\frac {24}{49}}(33sin({\frac {\pi }{7}})-10sin({\frac {2\pi }{7}})-6sin({\frac {3\pi }{7}}))\pi ={\frac {6}{7}}{\frac {4}{\sqrt {7}}}(3cos({\frac {\pi }{7}})+8cos({\frac {2\pi }{7}})-2cos({\frac {3\pi }{7}})-7)\pi }$ [7] ([8])

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {8n+7}{7}}}={\frac {7}{4}}(26log(2)+7{\sqrt {2}}log(3-2{\sqrt {2}}))}$ [9]

## Even d

 ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}}$ ${\displaystyle ={\frac {2}{3}}\cot \left({\frac {\pi }{3}}\right)\pi }$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {5n+4}{4}}}}$ ${\displaystyle ={\frac {4}{5}}\left(\cot \left({\frac {\pi }{5}}\right)-3\cot \left({\frac {2\pi }{5}}\right)\right)\pi }$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{\binom {7n+6}{6}}}}$ ${\displaystyle ={\frac {6}{7}}\left(\cot \left({\frac {\pi }{7}}\right)-5\cot \left({\frac {2\pi }{7}}\right)+10\cot \left({\frac {3\pi }{7}}\right)\right)\pi }$

For even values of ${\displaystyle d}$, a general expression in terms of alternating sign binomial coefficients and cotangent values follows from the identity

${\displaystyle \sum _{n=0}^{\infty }\left({\frac {1}{Ln+D}}-{\frac {1}{Ln+(L-D)}}\right)={\frac {\pi }{L}}\cot({\frac {D\pi }{L}})}$case D=1

(General expression for odd values of ${\displaystyle d}$?)

## Notes on the digamma function

${\displaystyle \Psi (1-z)-\Psi (z)=\pi \,\!\cot {\left(\pi z\right)}}$[10],6.3.7. in [11], 6.4.7. in [12]
${\displaystyle \Psi (1/2+z)-\Psi (1/2-z)=\pi \,\!\tan {\left(\pi z\right)}}$[13] ${\displaystyle =2\Psi (1+2z)-\Psi (1+z)+\Psi (1-z)-2\Psi (1-2z)\,}$[14]
${\displaystyle \left[\Psi (1-z)-\Psi (z)\right]\left[\Psi (1/2+z)-\Psi (1/2-z)\right]=\pi ^{2}=\left[\Gamma \left({\frac {1}{2}}\right)\right]^{4}}$

## Trigonometric functions defined from ${\displaystyle \Gamma (z)}$

${\displaystyle \sin(\pi z)={\frac {\Gamma ({\frac {1}{2}})^{2}}{\Gamma (1-z)\Gamma (z)}}}$[15]

${\displaystyle \cos(\pi z)={\frac {\Gamma ({\frac {1}{2}})^{2}}{\Gamma ({\frac {1}{2}}+z)\Gamma ({\frac {1}{2}}-z)}}}$[16][17] ${\displaystyle ={\frac {\Gamma (1-z)\Gamma (z)}{2\Gamma (1-2z)\Gamma (2z)}}}$[18] ${\displaystyle ={\frac {\Gamma (1-z)\Gamma (1+z)}{\Gamma (1-2z)\Gamma (1+2z)}}}$ [19]

${\displaystyle \tan(\pi z)={\frac {\Gamma ({\frac {1}{2}}+z)\Gamma ({\frac {1}{2}}-z)}{\Gamma (1-z)\Gamma (z)}}}$ [20]

### Sinc function

${\displaystyle {\frac {\sin(\pi z)}{\pi z}}={\frac {1}{\Gamma (1+z)\Gamma (1-z)}}}$[21][22]