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S = Σ [n=1→∞] (4n² - 1)/n!

S = Σ [n=1→∞] (4.n(n-1) + 4n -1)/n!

S = 4.Σ [n=1→∞] n(n-1)/n! + 4.Σ [n=1→∞] n/n! - Σ [n=1→∞] 1/n!

Let's trim the initial number of the sum, using n(n-1)=0 for n=1 and 1/n! = 1 for n=0

S = 4.Σ [n=2→∞] n(n-1)/n! + 4.Σ [n=1→∞] n/n! - Σ [n=0→∞] 1/n! + 1

We may now simplify and start counting from zero:

S = 4.Σ [n=2→∞] 1/(n-2)! + 4.Σ [n=1→∞] 1/(n-1)! - Σ [n=0→∞] 1/n! +1

S = 4.Σ [p=0→∞] 1/p! + 4.Σ [q=0→∞] 1/q! - Σ [n=0→∞] 1/n! +1

S = 7.Σ [n=0→∞] 1/n! + 1

S = 7e + 1