What comes next?

[Thomas Klemm]

(10-10-2024 09:17 PM)Gil Wrote:  Interesting, but I do not understand where the Matrix in your example comes from.

Just write the sequence A131689 downwards as an infinite triangular matrix:

\(
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\
& 1 & 1 & 1 & 1 & 1 & 1 & 1 &\cdots \\
& & 2 & 6 & 14 & 30 & 62 & 126 & \cdots \\
& & & 6 & 36 & 150 & 540 & 1806 & \cdots \\
& & & & 24 & 240 & 1560 & 8400 & \cdots \\
& & & & & 120 & 1800 & 16800 & \cdots \\
& & & & & & 720 & 15120 & \cdots \\
& & & & & & & 5040 & \cdots \\
& & & & & & & & \cdots \\
\end{bmatrix}
\)

There is a relative simple formula: \(T(n,k) = k \cdot \left(T(n-1,k-1) + T(n-1,k)\right)\) with \(T(0,0)=1\).

Examples

\(
\begin{align}
30 &= 2 \cdot (1 + 14) \\
1806 & = 3 \cdot (62 + 540) \\
8400 & = 4 \cdot (540 + 1560) \\
\end{align}
\)