Product Formula — ∏G(k) = (n!)²/(n+1)ⁿ

∏_k=1ⁿ G(k) = (n!)² / (n+1)ⁿ

Compute Product

N = 10

Product Growth

Step-by-Step Accumulation

k	G(k)	∏G(1..k)	(k!)²/(k+1)^k	Match

The Proof

G(k) = k^k+1/(k+1)^k

∏G(k) = ∏ k^k+1/(k+1)^k

Telescope: numerator = ∏k^k+1 = 1²·2³·3⁴···nⁿ⁺¹
denominator = ∏(k+1)^k = 2¹·3²·4³···(n+1)ⁿ

After cancellation: (n!)² / (n+1)ⁿ ∎