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\noindent\bf Formula \rm

 The formula for the coefficient in the case where $s= d-1$ and the 
monomial $M$ has a matrix of 
exponents  
 $$(d-1) \left[{\matrix{3&0&1
 \cr 1&3&0 \cr 0&1&3 \cr}}\right] ,$$ 
is given as $A = A_1 +  A_2 + A_3 + A_4 + A_5 + A_6$, where the $A_i$ are given 
as follows:
$$ A_1 = \sum _{l=0} ^s \sum _{j=0} ^l \sum _{i=0} ^{s-l} {s \choose l}^3 {l\choose j}^2 \left( {l\choose j+1} - {l\choose j} \right) {s-l\choose i}^2 \left( {s-l\choose i+1} - {s-l\choose i} \right) .
$$
$$ \eqalign{ A_2 = \sum _{l=0} ^{s-1}  \sum _{j=0} ^l  
{s \choose l}^2 & {s \choose l+1} {l\choose j} \left( {l\choose j+1} {l+1\choose j+2} + {l\choose j+1} {l+1\choose j+1} - 2 {l\choose j} {l+1\choose j+1}
\right)\cr & \sum _{i=0} ^{s-l}{s-l \choose i} \left( {s-l\choose i}  {s-l-1 \choose i-1} - {s-l \choose i+1} {s-l -1\choose i}\right) .\cr}
$$
$$ \eqalign{ A_3 = \sum _{l=0} ^{s-2}  \sum _{j=0} ^l 
{s \choose l}^2 & {s \choose l+2} 
{l\choose j} \left( {l\choose j+1} {l+2\choose j+2} - {l\choose j} {l+2\choose j+1}\right) \cr & 
\sum _{i=0} ^{s-l}{s-l \choose i}  {s-l-2\choose i-1}  \left( {s-l \choose i+1} - {s-l \choose i}\right) .\cr}$$
$$ \eqalign{ A_4 = \sum _{l=0} ^{s-1}  \sum _{j=0} ^{l +1}
{s \choose l+1}^2 & {s \choose l} 
 \left( {l\choose j+1} {l+1\choose j+1} + {l\choose j} {l+1\choose j+1} - 2 {l+1\choose j} {l \choose j-1 }\right) \cr & {l+1 \choose j}\sum _{i=0} ^{s-l-1}{s-l-1 \choose i}  {s-l\choose i+1}  \left( {s-l-1 \choose i} - {s-l-1 \choose i+1}\right) .\cr}$$
$$ \eqalign{ A_5 = \sum _{l=0} ^{s-2}  \sum _{j=0} ^{l}
& {s \choose l} {s \choose l+1} {s \choose l+2}
 {l +1\choose j}  \left( {l\choose j+1} {l+2\choose j+2} -  {l\choose j} {l+2  \choose j+1 }\right) \cr & \sum _{i=0} ^{s-l-1} {s-l-1 \choose i}  {s-l\choose i+1}  \left( {s-l-2 \choose i} - {s-l-2 \choose i-1}\right) .\cr}$$
$$ \eqalign{ A_6 = \sum _{l=0} ^{s-2}  \sum _{j=1} ^{l+2}
{s \choose l+2}^2 & {s \choose l+2}
 {l +2\choose j}  \left( {l\choose j-2} {l+2\choose j-1} -  {l\choose j-1} {l+2  \choose j }\right) \cr & \sum _{i=0} ^{s-l-2} {s-l-2 \choose i}  {s-l\choose i+1}  \left( {s-l-2 \choose i-1} - {s-l-2 \choose i}\right) .\cr}$$

\end
