设 S = {1,2,3,..., n}, 令 f(S) 表示所有 S 的非空子集和的平方和, 即:...

Summing squares of subset sums

OK, let's think this through: we're calculating the sum of squares of sums for all non-empty subsets of a set $S$, and then take it modulo $10^9 + 7$.

Breaking it down

I’m working through the identity for $f(S)$ by expanding and separating sums, examining each $x$ in $T$, and considering all non-empty subsets $T$.

Confirming subset contributions

I’m verifying that each element $x$ in $S$ appears in exactly $2^{n-1}$ non-empty subsets, ensuring the sum of $x^2$ across all $T$ accurately reflects these contributions.

Summing pairs

I’m gathering info on the sum of products $x$ and $y$ over all non-empty subsets $T$, considering how each distinct pair appears in $2^{n-2}$ subsets.

Summing distinct pairs

I’m adding up the products of all distinct pairs $x$ and $y$ from $\{1, \ldots, n\}$, ensuring each pair is counted only once in the sum.

Crafting the formula

I’m working on simplifying $f(S)$ by unifying the sums of squares for each $x$ in $S$ with the sum of products for distinct pairs $(x, y)$, leading to a concise expression.

Breaking down the approach

I’m simplifying the expression by noting that the sum of squares plus the sum of products for distinct pairs is half the sum of the squares plus the square of the sum of $x$.

Combining and simplifying

I’m amalgamating terms, isolating common factors, and streamlining the algebraic expression to deduce a simpler closed-form solution for $f(S)$.

Finalizing the expression

I’m adding the final touches to a simplified version of $f(S)$, factoring $3n^2+7n+2$ into $(3n+1)(n+2)$, and ensuring all terms are neatly combined for a clearer form.

Finalizing the steps

I’m mapping out the calculation for $f(S)$ modulo $10^9 + 7$ by carefully breaking down the terms, ensuring each part is accurately computed and combined to find the final result.

Isolating powers

I’m piecing together the factorization of large numbers to simplify $f(S)$ modulo $10^9 + 7$, isolating common factors and simplifying the division by 12.

Breaking down calculations

I’m meticulously assembling the factorization of large numbers to streamline $f(S)$ modulo $10^9 + 7$, isolating common factors, and making the division by 12 easier.

Calculating powers

I’m multiplying prime factors of large numbers to simplify $f(S)$ modulo $10^9 + 7$, accumulating powers of 2, 3, 5, 7, 17, 11, 23, and 607 to streamline the division by 12.

Pulling together

OK, let me see. I’m combining $P$ divided by 12 with $2^{2021}$ to form $Q$, then aligning these terms for $f(S)$ modulo $10^9 + 7$.

Ensuring precision

I’m moving through the steps to compute $f(S)$ modulo $10^9 + 7$, ensuring each term is accurately factored and multiplied, while maintaining the integrity of the factors.

Breaking down the exponentiation

OK, let me see. I’m thinking about a method to compute $2^{2023}$ modulo $10^9 + 7$ by breaking it into smaller parts and using modular exponentiation techniques.