History of Butcher tableaux of explicit Runge–Kutta methods

 Title: Butcher tableaux of explicit Runge–Kutta methods-Definition: An $s$-stage explicit Runge–Kutta method advances $y'=f(t,y)$ by $y_{n+1}=y_n+h\sum_i-  b_i k_i$, where $k_i=f(t_n+c_i h, y_n+h\sum_{j<i} a_{ij}k_j)$. Its Butcher tableau-  CITE{WikiList} consists of the nodes $c_i$, the strictly lower triangular coefficients-  $a_{ij}$, and the weights $b_i$; for an embedded pair, the table records one weight-  row $b^{(q)}$ for each order $q$.+Definition: A Butcher tableau CITE{WikiList} for an explicit Runge–Kutta method records+  the nodes $c_i$, strictly lower triangular coefficients $a_{ij}$, and weights $b_i$+  in the one-step formula. An embedded pair has one weight row $b^{(q)}$ for each+  order $q$. Parameters:   method:

     fourth-order method, whose coefficients lie in $\mathbb{Q}(\sqrt5)$. Formulas:+  formula-step: The one-step formula is $y_{n+1}=y_n+h\sum_i b_i k_i$, where $k_i=f(t_n+c_i+    h, y_n+h\sum_{j<i} a_{ij}k_j)$; for an embedded pair, use the row $b^{(q)}$ in+    place of $b$.   formula-row-sums: For every stage, $c_i=\sum_{j<i}a_{ij}$.   formula-moments: For each order-$q$ weight row and $1\leq m\leq q$, $\sum_i b^{(q)}_i