is the smallest degree greater than 1 of the irreducible representations of the Baby Monster group. Define, () Then, if the series converges and the sign chosen appropriately. Examples: The first formula, found by Ramanujan and mentioned at the start of the article, belongs to a family proven by D. Bailey and the Borwein brothers in a 1989 paper. Level 3 Define, where 782 is the smallest degree greater than 1 of the irreducible representations of the Fischer group Fi23 and, () Examples: Level 4 Define, where the first is the 24th power of the Weber modular function . And, () () Examples: Level 5 Define, and, () where the first is the product of the central binomial coefficients and the Apéry numbers () Examples: Level 6 Modular functions In 2002, Sato established the first results for levels above 4. It involved Apéry numbers which were first used to establish the irrationality of . First, define, J. Conway and S. Norton showed there are linear relations between the McKay–Thompson series Tn, one of which was, or using the above eta quotients jn, α Sequences For the modular function j6A, one can associate it with three different sequences. (A similar situation happens for the level 10 function j10A.) Let, (, labeled as s6 in Cooper's paper) () The three sequences involve the product of the central binomial coefficients with: first, the Franel numbers ; second, , and third, . Note that the second sequence, α2(k) is also the number of 2n-step polygons on a cubic lattice. Their complements, There are also associated sequences, namely the Apéry numbers, () the Domb numbers (unsigned) or the number of 2n-step polygons on a diamond lattice, () and the Almkvist-Zudilin numbers, () where Identities The modular functions can be related as, if the series converges and the sign chosen appropriately. It can also be observed that, which implies, and similarly using α3 and α'3. Examples One can use a value for j6A in three ways. For example, starting with, and noting that then, as well as, though the formulas using the complements apparently do not yet have a rigorous proof. For the other modular functions, Level 7 Define () and, Example: No pi formula has yet been found using j7B. Level 8 Define, The expansion of the first is the McKay–Thompson series of class 4B (and is the square root of another function). The fourth