How can I create the following symbol?

PS: It is not the classic \sum_{i=0}^n k
How can I create the following symbol?

PS: It is not the classic \sum_{i=0}^n k
At \scriptscriptstyle the output is not satisfying, but I don't think you need it.
\documentclass{article}
\usepackage{amsmath}
\makeatletter
\def\mov@rlay#1#2{\leavevmode\vtop{%
\baselineskip\z@skip \lineskiplimit-\maxdimen
\ialign{\hfil$\m@th#1##$\hfil\cr#2\crcr}}}
\newcommand{\charfusion}[3][\mathord]{
#1{\ifx#1\mathop\vphantom{#2}\fi
\mathpalette\mov@rlay{#2\cr#3}
}
\ifx#1\mathop\expandafter\displaylimits\fi}
\newcommand{\sumk}{\charfusion[\mathop]{\sum}{\innerk}}
\newcommand{\innerk}{%
\mathchoice
{\mbox{\fontsize\f@size{0}\normalfont\ k}}
{\mbox{\fontsize\sf@size{0}\normalfont\ k}}
{\mbox{\fontsize\ssf@size{0}\normalfont\ k}}
{\mbox{\tiny\normalfont\ k}}%
}
\makeatother
\begin{document}
\[
\sumk\textstyle\sumk\scriptstyle\sumk\scriptscriptstyle\sumk
\]
\[
\sumk_{i=1}^{n} a_{i}
\]
\end{document}
(see my answer to Mathematical symbol for disjoint set union)

A different implementation, where the inner symbol is given as argument.
\documentclass{article}
\usepackage{amsmath}
\makeatletter
\newcommand{\sumk}[1]{\mathop{\vphantom{\sum}\mathpalette\sum@k{#1}}}
\newcommand{\sum@k}[2]{%
\ooalign{%
\sum@k@vc{#1}{\sum}\cr
\hidewidth\sum@k@vc{\sum@k@next{#1}}{\mkern8mu#2}\hidewidth\cr}%
}
\newcommand{\sum@k@vc}[2]{%
$\vcenter{\hbox{$\m@th#1#2$}}$%
}
\newcommand{\sum@k@next}[1]{%
\ifx#1\displaystyle\displaystyle\fi
\ifx#1\textstyle\scriptstyle\fi
\ifx#1\scriptstyle\scriptscriptstyle\fi
\ifx#1\scriptscriptstyle\scriptscriptstyle\fi
}
\makeatother
\begin{document}
\[
\sumk{k}_{i=1}^n\textstyle\sumk{h}\scriptstyle\sumk{r}
\]
\end{document}

% arara: pdflatex
\documentclass{article}
%\newcommand{\inSum}[1]{\mkern-15mu #1} % if you use that often, you can write \sum_{i=0}^{n}\inSum{j} or alike. Works only in in display math.
\begin{document}
\begin{equation}
\sum_{i=0}^{n}\mkern-16mu \mathrm{k} = \dots
\end{equation}
\end{document}
The \mathrm{} sets the letter in upright style. You may want to take this away, as k can vary and thus is a variable (italic).
Play around with the -16mu to other values, until you like it.

This makes an operator that will do what you wish, with a parameterized style that allows any letter to be put inside the summation:
\documentclass{article}
\usepackage{amsopn}
\newcommand{\suminside}[1]{\operatorname*{\sum \mkern-14mu \makebox[0.77em][l]{\footnotesize #1}}}
\begin{document}
\begin{equation}
\suminside{k}_{i=0}^n a_i = 1
\end{equation}
\begin{equation}
\suminside{d}_{i=0}^n a_i = 2
\end{equation}
\centering
Inline displays as \( \suminside{k}_{i=0}^n a_i = 2 \).
\end{document}
appears as:

\usepackage{mathtools}, you can do\mathrlap{\sum_{i=0}^n} \;\;\; \text{\scriptsize k}. – Aung Mar 09 '14 at 00:31