ベルヌーイ数

母関数

zez1=n=0Bnn!zn(|z|<2π)\frac{z}{e^z-1}=\sum_{n=0}^\infty \frac{B_n}{n!}z^n \qquad (\lvert z\rvert<2\pi)

漸化式

B0=1,Bn=1n+1k=0n1(n+1k)Bk(n1)\begin{aligned} B_0&=1, \\ B_n&=-\frac{1}{n+1}\sum_{k=0}^{n-1}\binom{n+1}{k}B_k \qquad (n\ge1)\end{aligned}

B0=1B1=12B2=16B3=0B4=130B5=0B6=142B7=0B8=130B9=0B10=566B11=0B12=6912730B13=0B14=76B15=0B16=3617510B17=0B18=43867798B19=0B20=174611330B21=0B22=854513138B23=0B24=2363640912730B25=0B26=85531036B27=0B28=23749461029870B29=0\begin{aligned} B_{0}&={1} \\ B_{1}&={-{\frac{1}{2}}} \\ B_{2}&={\frac{1}{6}} \\ B_{3}&={0} \\ B_{4}&={-{\frac{1}{30}}} \\ B_{5}&={0} \\ B_{6}&={\frac{1}{42}} \\ B_{7}&={0} \\ B_{8}&={-{\frac{1}{30}}} \\ B_{9}&={0} \\ B_{10}&={\frac{5}{66}} \\ B_{11}&={0} \\ B_{12}&={-{\frac{691}{2730}}} \\ B_{13}&={0} \\ B_{14}&={\frac{7}{6}} \\ B_{15}&={0} \\ B_{16}&={-{\frac{3617}{510}}} \\ B_{17}&={0} \\ B_{18}&={\frac{43867}{798}} \\ B_{19}&={0} \\ B_{20}&={-{\frac{174611}{330}}} \\ B_{21}&={0} \\ B_{22}&={\frac{854513}{138}} \\ B_{23}&={0} \\ B_{24}&={-{\frac{236364091}{2730}}} \\ B_{25}&={0} \\ B_{26}&={\frac{8553103}{6}} \\ B_{27}&={0} \\ B_{28}&={-{\frac{23749461029}{870}}} \\ B_{29}&={0}\end{aligned}


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