A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions
Abstract
:1. Introduction and the Main Recursion Formulas
2. An Explicit Formula for the Numbers
3. The -Partition Function
- (1)
- : .
- (2)
- : Based on the results presented in Table 2, we observe that the impossible partitions for are obtained by adding the number 1 to the impossible partitions for . The total number of such partitions is given by . In addition to these, we also consider the impossible partitions and , whose number is given by . Therefore, we find thatWe now formulate the following generalization:
- (3)
- : Analogously, we have number of partitions , , + number of partitions . We then haveFor , the impossible partition is already included in the count of . Therefore, we can express the total number of impossible partitions for as follows:This leads to the generalization given by
- (4)
- : In this case, in addition to the partitions and , we also consider the partition . Thus, clearly, we can write the total number of impossible partitions as , so that
- (5)
- : For , the number of impossible partitions of the formFor , the impossible partition is already included in the count of . Therefore, we can express the total number of impossible partitions for as follows:For , the expression for is given byFor , we adjust the formula as follows:In general, we can express as follows:
- (6)
- : For , the number of impossible partitions of the formFor , we obtain the following expression for :For , the expression for is given byFor , we haveFor , we observe that certain partitions are double-counted. Specifically, the partitions of the following form:In this procedure, we mark the number of impossible partitions that are repeated for the number with —that is,Thus, in general, we can express as follows:
4. Connection of the Eisenstein Series with the Numbers
5. Decomposition of the Numbers
6. Concluding Remarks and Observations
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
1 | 2 | 2 | 4 | 5 | 7 | 9 | 13 | |
1 | 2 | 3 | 4 | 6 | 9 | 12 | 16 | |
1 | 2 | 3 | 5 | 6 | 10 | 13 | 18 |
Impossible Partitions | ||
---|---|---|
25 | 1 | |
26 | 1 | |
27 | , | 2 |
28 | , , | 3 |
29 | , , , , | 5 |
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Petojević, A.; Srivastava, H.M.; Orlić, S. A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions. Axioms 2025, 14, 451. https://doi.org/10.3390/axioms14060451
Petojević A, Srivastava HM, Orlić S. A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions. Axioms. 2025; 14(6):451. https://doi.org/10.3390/axioms14060451
Chicago/Turabian StylePetojević, Aleksandar, Hari M. Srivastava, and Sonja Orlić. 2025. "A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions" Axioms 14, no. 6: 451. https://doi.org/10.3390/axioms14060451
APA StylePetojević, A., Srivastava, H. M., & Orlić, S. (2025). A Class of the Generalized Ramanujan Tau Numbers and Their Associated Partition Functions. Axioms, 14(6), 451. https://doi.org/10.3390/axioms14060451