  For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler
$T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual
infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd
denominators. Like the Euler sums, the Euler $T$-sums can be evaluated
according to the Contour integral and residue theorem. Using this fact, we
obtain explicit formulas for Euler $T$-sums with repeated arguments analogous
to those known for Euler sums. Euler $T$-sums can be written as rational linear
combinations of the Hoffman $t$-values. Using known results for Hoffman
$t$-values, we obtain some examples of Euler $T$-sums in terms of (alternating)
multiple zeta values. Moreover, we prove an explicit formula of triple
$t$-values in terms of zeta values, double zeta values and double $t$-values.
We also define alternating Euler $T$-sums and prove some results about them by
the Contour integral and residue theorem. Furthermore, we define another Euler
type $T$-sums and find many interesting results. In particular, we give an
explicit formulas of triple Kaneko-Tsumura $T$-values of even weight in terms
of single and the double $T$-values. Finally, we prove a duality formula of
Kaneko-Tsumura's conjecture.
