Recurrent formula of Bernoulli numbers and the relationships among the coefficients of beam, Bernoulli numbers and Euler numbers

Based on the differential equation of the deflection curve for the beam, the equation of the deflection curve for the simple beam is obtained by integral. The equation of the deflection curve for the simple beam carrying the linear load is generalized, and then it is expanded into the corresponding Fourier series. With the obtained summation results of the infinite series, it is found that they are related to Bernoulli numbers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam, Bernoulli numbers and Euler numbers are found, and the relative mathematical formulas are presented.