4,2

Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]

$\dsum\limits_{k_{2}=0}^{\lfloor \frac{n}{2}\rfloor }\dsum\limits_{h=0}^{n}\dsum\limits_{f=0}^{n-h}\dsum\limits_{b_{1}=0}^{% \lfloor \frac{f+2h-k_{2}}{2}\rfloor }\dsum\limits_{k_{4}=0}^{\min \{\lfloor \frac{n-2k_{2}}{4}\rfloor ,(n-f-h)\}}\dsum\limits_{k_{3}=0}^{\min \{\lfloor \frac{4n-2f-4h-4k_{4}}{3}\rfloor ,\lfloor \frac{n-2k_{2}-4k_{4}}{3}\rfloor ,(n-f-h-k_{4})\}}\dsum\limits_{k_{1}=0}^{\min \{(4n-2f-4h-3k_{3}-4k_{4}),(n-2k_{2}-3k_{3}-4k_{4})\}}\bigskip \dsum\limits_{d=0}^{\min \{\lfloor \frac{4n-2f-4h-k_{1}-3k_{3}-4k_{4}}{8}% \rfloor ,(n-k_{3}-k_{4}-f-h)\}}\dsum\limits_{c=0}^{\min \{\lfloor \frac{% 4n-2f-4h-8d-k_{1}-3k_{3}-4k_{4}}{6}\rfloor ,\lfloor \frac{% n-2d-k_{3}-k_{4}-f-h}{2}\rfloor

\}}\dsum\limits_{b_{2}=0}^{\min \{\lfloor \frac{4n-2f-4h-6c-8d-k_{1}-3k_{3}-4k_{4}}{4}\rfloor ,(n-2c-2d-k_{3}-k_{4}-f-h)\}}\dsum\limits_{a_{1}=\lceil \frac{% (4n-2f-4h-4b_{2}-6c-8d-k_{1}-3k_{3}-4k_{4})}{2}\rceil }^{\lfloor \frac{% (4n-2f-4h-4b_{2}-6c-8d-k_{1}-3k_{3}-4k_{4})}{2}\rfloor }$\bigskip ($\frac{% (-1)^{(f+2h-2b_{1}-k_{2})+b_{1}+d+k_{2}+k_{4}}}{% 2^{3n+a_{1}-b_{2}-5c-3d-3k_{3}-k_{4}-f}3^{n-b_{2}-c-2d-k_{4}-f-h}}\times \frac{n!((f+2h-2b_{1}-k_{2})+2b_{1}+k_{2})!(2a_{1}+k_{1})!}{% a_{1}!(f+2h-2b_{1}-k_{2})!b_{1}!b_{2}!c!d!k_{1}!k_{2}!k_{3}!k_{4}!f!h!(n-b_{2}-2c-2d-k_{3}-k_{4}-f-h)!% })$ [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]

$\sum_{2a_{1}+2a_{2}+4b_{1}+4b_{2}+6c+8d+e_{1}+2e_{2}+3e_{3}+4e_{4}=4n}% \sum_{f+2h=a_{2}+2b_{1}+e_{2}}\frac{(-1)^{a_{2}+b_{1}+d+e_{2}+e_{4}}}{% 2^{3n+a_{1}-b_{2}-5c-3d-3e_{3}-e_{4}-f}3^{n-b_{2}-c-2d-e_{4}-f-h}}\times \frac{n!(a_{2}+2b_{1}+e_{2})!(2a_{1}+e_{1})!}{% a_{1}!a_{2}!b_{1}!b_{2}!c!d!e_{1}!e_{2}!e_{3}!e_{4}!f!h!(n-b_{2}-2c-2d-e_{3}-e_{4}-f-h)!% },$where the sum is over all nonnegative solutions of $% 2a_{1}+2a_{2}+4b_{1}+4b_{2}+6c+8d+e_{1}+2e_{2}+3e_{3}+4e_{4}=4n$, $% f+2h=a_{2}+2b_{1}+e_{2}$, and $e_{1}+2e_{2}+3e_{3}+4e_{4}\leq n$. [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]

a(4) = 1:

1111

1111

1111

1111

Cf. A000085 (row sums 1), A000986 (row sums 2), A110040 (row sums 3).

Sequence in context: A143900 A091742 A160140 this_sequence A160261 A037138 A091429

Adjacent sequences: A139667 A139668 A139669 this_sequence A139671 A139672 A139673

nonn

Ron Hardin (rhhardin(AT)att.net), Jun 12 2008