P5627 [AFOI-19] sum and prod

SY finally tidied up her messy quilt. As soon as she arrived in the classroom, she received a note from QM... To: Dear SY     Take a look at the formula I dreamed of last night. Solve it and I will give you candy. From: Your QM. SY, of course, could not resist the temptation of $C_{6}H_{12}O_{6}$. But when she saw the fancy formula on the back of the note, she was stunned... Still, SY really wanted to eat candy.

Find the value of $$\sum_{i=1}^{2^{n}}\log_{2}{(\prod_{j = 1}^{i}lowbit(j))}$$ where $lowbit(x)$ means the result of ` x&(~x+1)`.

One line containing an integer $n$.

One line containing one integer, the answer modulo $10^9+7$.

For the first $20\%$ of the testdata, $1 \leq n \leq 60$. For the first $50\%$ of the testdata, $1 \leq n \leq 10^4$. For the full $100\%$ of the testdata, $1 \leq n \leq 2^{62}$. Translated by ChatGPT 5