Geodesic
Euler’s “exemplum memorabile inductionis fallacis” and 𝑞-trinomial coefficients
Journal of the American Mathematical Society, Tome 03 (1990) no. 3, pp. 653-669

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The trinomial coefficients are defined centrally by $\Sigma _{j = - m}^\infty {(_j^m)_2}{x^j} = {(1 + x + {x^{ - 1}})^m}$. Euler observed that for $- 1 \leq m \leq 7$, $3{(_{ \;0}^{m + 1})_2} - {(_{ \;0}^{m + 2})_2} = {F_m}({F_m} + 1)$, where ${F_m}$ is the $m$th Fibonacci number. The assertion is false for $m > 7$. We prove general identities—one of which reduces to Euler’s assertion for $m \leq 7$. Our main object is to analyze $q$-analogs extending Euler’s observation. Among other things we are led to finite versions of dissections of the Rogers-Ramanujan identities into even and odd parts. 
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Andrews, George E. Euler’s “exemplum memorabile inductionis fallacis” and 𝑞-trinomial coefficients. Journal of the American Mathematical Society, Tome 03 (1990) no. 3, pp. 653-669. doi: 10.1090/S0894-0347-1990-1040390-4

[1] Andrews, George E. Sieves in the theory of partitions Amer. J. Math. 1972 1214 1230

[2] Andrews, George E. The theory of partitions 1976

[3] Andrews, George E. The hard-hexagon model and Rogers-Ramanujan type identities Proc. Nat. Acad. Sci. U.S.A. 1981 5290 5292

[4] Andrews, George E. Use and extension of Frobenius’ representation of partitions 1984 51 65

[5] Andrews, George E., Baxter, R. J. Lattice gas generalization of the hard hexagon model. III. 𝑞-trinomial coefficients J. Statist. Phys. 1987 297 330

[6] Computer algebra 1989

[7] Andrews, George E., Baxter, R. J., Bressoud, D. M., Burge, W. H., Forrester, P. J., Viennot, G. Partitions with prescribed hook differences European J. Combin. 1987 341 350

[8] Andrews, George E., Baxter, R. J., Forrester, P. J. Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities J. Statist. Phys. 1984 193 266

[9] Bressoud, David M. Extension of the partition sieve J. Number Theory 1980 87 100

[10] Burge, William H. A correspondence between partitions related to generalizations of the Rogers-Ramanujan identities Discrete Math. 1981 9 15

[11] Comtet, Louis Advanced combinatorics 1974

[12] Slater, L. J. Further identities of the Rogers-Ramanujan type Proc. London Math. Soc. (2) 1952 147 167

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