Maclaurin series: $\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^7}{7!}+\frac{x^8}{8!}+\frac{x^{11}}{11!}+\frac{x^{12}}{12!}+...$ The Taylor series of a real or complex-valued function ƒ(x) that is infinitely differentiable at a real or complex number a is the power series
$$f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. $$
I have a curiosity concerning the following polynomial:
$$\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^7}{7!}+\frac{x^8}{8!}+\frac{x^{11}}{11!}+\frac{x^{12}}{12!}+...$$
Is this a known development in Maclaurin series?

Note that 
$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots.\tag{1}$$
A little less familiar is 
$$\sinh x=\frac{e^x-e^{-x}}{2}=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots.\tag{2}$$
Subtract (1) from (2) and divide by $2$. We get
$$\frac{1}{2}(\sinh x-\sin x)=\frac{x^3}{3!}+\frac{x^7}{7!}+\frac{x^{11}}{11!}+\cdots.$$
That takes care of the odd guys. Now play a similar game with $\cos x$ and $\cosh x$. Or integrate the closed form  for the sum of the odds from $0$ to $x$.
