A Certain Unscientific Polynomial-Weighted Geometric Sum with Updates — Alternative Solution¶
Codex (GPT-6) derived and implemented the moment recurrences and constant-time coefficient updates for this problem. See alternative.cpp.
Store all moments¶
For fixed current \(n\) , define \(S_j(n)=\sum_{i=1}^n a^i i^j\) . The answer is \(\sum_jb_jS_j(n)\) . A single coefficient update adds \(\Delta b_jS_j(n)\) in \(O(1)\) time. Recompute moments only when the upper bound changes.
For \(a\ne1\) , subtract shifted sums and expand \((i-1)^j\) to obtain
In particular, \(S_0=(a-a^{n+1})/(1-a)\) . Compute moments in increasing degree; do not omit the boundary term \(a[j=0]\) . The same formula works for \(a=0\) .
For \(a=1\) , telescope \((i+1)^{j+1}-i^{j+1}\) to get
Use precomputed modular inverses for division and multiplication to advance consecutive powers \(n^j\) and \((n+1)^{j+1}\) .
Correctness, Complexity, and Status¶
Both recurrences follow by finite-sum identities and depend only on lower moments. Induction proves every \(S_j\) correct, and hence the inner product and incremental coefficient update are correct.
Let \(q\) be the number of upper-bound updates. Binomial preprocessing takes \(O(k^2)\) ; each recomputation takes \(O(k^2+\log n+\log P)\) ; coefficient updates cost \(O(1)\) ; space is \(O(k^2)\) . Coefficient-heavy workloads avoid repeatedly maintaining a whole polynomial, but exclusively changing bounds gives \(O(mk^2)\) work. The slowest five-run formal median was about 0.702 seconds, above the 0.5-second limit. This is a backup; use the triangular-transform method for the complete range.