\(\qquad \qquad\displaystyle I_n(x) = \frac{2^n}{\sqrt{\pi}}\int\limits_{-\infty}^{\infty}\frac{(x+it)^n}{e^{t^2}} dt\)
###### This is a part of Hot Integrals

Then, the following summation can be expressed as:

\(\displaystyle \sum_{n=0}^{\infty} \frac{I_n(x)2^n}{\lfloor \frac{n}{2}\rfloor !} = \frac{a+bx}{a^{\frac{c}{d}}}e^{\frac{p}{a}x^q}\)

Calculate \(a+b+c+d+p+q\).

**Details and Assumptions**

1)\(i=\sqrt{-1}\)

2) \(a,b,c,d,p,q\) are positive integers.

3) \(\gcd(c,d)=\gcd(p,a)=1\)

4)Sign \(!\) indicates factorial. For e.g. \(3! = 3 \times 2 \times 1\)

5)\( \lfloor . \rfloor \) represents the floor function.

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