Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908408839405568 |
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| author | Tang, Jinlong Xin, Guoce |
| author_facet | Tang, Jinlong Xin, Guoce |
| contents | This paper resolves an open problem raised by Ekhad and Zeilberger for computing $ω(10000)$, which is related to Stern's triangle. While $ν(n)$, defined as the sum of squared coefficients in $\prod_{i=0}^{n-1} (1 + x^{2^i} + x^{2^{i+1}})$, admits a rational generating function, the analogous function $ω(n)$ for $\prod_{i=0}^{n-1} (1 + x^{2^i+1} + x^{2^{i+1}+1})$ presents substantial computational difficulties due to its complex structure.
We develop a method integrating constant term techniques, conditional transfer matrices, algebraic generating functions, and $P$-recursions. Using the conditional transfer matrix method, we represent $ω(n)$ as the constant term of a bivariate rational function. This framework enables the calculation of $ω(10000)$, a $6591$-digit number, and illustrates the method's broad applicability to combinatorial generating functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle Tang, Jinlong Xin, Guoce Combinatorics Primary 05A15, Secondary 05A10 This paper resolves an open problem raised by Ekhad and Zeilberger for computing $ω(10000)$, which is related to Stern's triangle. While $ν(n)$, defined as the sum of squared coefficients in $\prod_{i=0}^{n-1} (1 + x^{2^i} + x^{2^{i+1}})$, admits a rational generating function, the analogous function $ω(n)$ for $\prod_{i=0}^{n-1} (1 + x^{2^i+1} + x^{2^{i+1}+1})$ presents substantial computational difficulties due to its complex structure. We develop a method integrating constant term techniques, conditional transfer matrices, algebraic generating functions, and $P$-recursions. Using the conditional transfer matrix method, we represent $ω(n)$ as the constant term of a bivariate rational function. This framework enables the calculation of $ω(10000)$, a $6591$-digit number, and illustrates the method's broad applicability to combinatorial generating functions. |
| title | Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle |
| topic | Combinatorics Primary 05A15, Secondary 05A10 |
| url | https://arxiv.org/abs/2506.13375 |