Unknown Catalan Number series!

$\underline{\color{#007fff}{\textbf{Method-1}}}$

The first five Catalan numbers for $n = 1, 2, 3 ,4,5$ are $1, 2, 5, 14, 42$ .

Using Polynomial Interpolation , $F(x)=\dfrac{42(x-1)(x-2)(x-3)(x-4)}{24}-\dfrac{14(x-1)(x-2)(x-3)(x-5)}{6}+\dfrac{5(x-1)(x-2)(x-4)(x-5)}{4}-\dfrac{2(x-1)(x-3)(x-4)(x-5)}{6}+\dfrac{(x-2)(x-3)(x-4)(x-5)}{24}$

Putting $x=6$ and $7$ respectively, we get $F(6)=111,F(7)=252$ . Hence $F(7)+F(6)=252+111=\boxed{\color{#0C6AC7}{363}}$ .

---

$\underline{\color{#007fff}{\textbf{Method-2}}}$

Using Method of Differences ,

$\begin{array}{c|c|c|c|c|c} n & f(n) & D_{1} & D_{2} & D_{3} & D_{4} \\\hline1 & 1 & 1& 2 & 4 & \color{#3D99F6}{9}\\\hline 2 & 2 & 3 & 6 & 13 & \color{#3D99F6}{9}\\\hline 3 & 5 & 9 & 19 & 22 & \color{#3D99F6}{9}\\\hline 4 & 14 & 28 & 41 & 31 & \color{#3D99F6}{9}\\\hline 5 & 42 & 69 & 72 & & \color{#3D99F6}{9}\\\hline \color{#D61F06}{6} & \color{#D61F06}{111} & 141 & & & \color{#3D99F6}{9} \\\hline \color{#20A900}{7} & \color{#20A900}{252} & & & & \color{#3D99F6}{9} \\\hline \end{array}$

Hence,

$\color{#D61F06}{f(6)}+\color{#20A900}{f(7)}=\color{#D61F06}{111}+\color{#20A900}{252}=\boxed{\color{#0C6AC7}{363}}$