Let $T(n, m)$ be the number of $m$-tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$.
For example, $T(3, 4)=8$, via the following eight $4$-tuples:
$(1, 1, 1, 1)$
$(1, 1, 1, 2)$
$(1, 1, 2, 1)$
$(1, 2, 1, 1)$
$(1, 2, 1, 2)$
$(2, 1, 1, 1)$
$(2, 1, 1, 2)$
$(2, 1, 2, 1)$
You are also given that $T(5, 5)=246$, $T(10, 10^{2}) \equiv 862820094 \pmod{1\,000\,000\,007}$ and $T(10^2, 10) \equiv 782136797 \pmod{1\,000\,000\,007}$.
Find $T(5000, 10^{12}) \bmod 1\,000\,000\,007$.
Solution
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