is also the square root of another function. Let, where the first is the product of the central binomial coefficient and a sequence related to an arithmetic-geometric mean (), Examples: though no pi formula is yet known using j8A(τ). Level 9 Define, The expansion of the first is the McKay–Thompson series of class 3C (and related to the cube root of the j-function), while the second is that of class 9A. Let, where the first is the product of the central binomial coefficients and (though with different signs). Examples: Level 10 Modular functions Define, Just like the level 6, there are also linear relations between these, or using the above eta quotients jn, β sequences Let, (, labeled as s10 in Cooper's paper) their complements, and, though closed forms are not yet known for the last three sequences. Identities The modular functions can be related as, if the series converges. In fact, it can also be observed that, Since the exponent has a fractional part, the sign of the square root must be chosen appropriately though it is less an issue when jn is positive. Examples Just like level 6, the level 10 function j10A can be used in three ways. Starting with, and noting that then, as well as, though the ones using the complements do not yet have a rigorous proof. A conjectured formula using one of the last three sequences is, which implies there might be examples for all sequences of level 10. Level 11 Define the McKay–Thompson series of class 11A, where, and, No closed form in terms of binomial coefficients is yet known for the sequence but it obeys the recurrence relation, with initial conditions s(0) = 1, s(1) = 4. Example: Higher levels As pointed out by Cooper, there are analogous sequences for certain higher levels. Similar series R. Steiner found examples using Catalan numbers , and for this a modular form with a second periodic for k exists: Other similar series are with the last (comments in ) found by using a linear combination of higher parts of Wallis-Lambert series for and Euler series for the circumference of an ellipse. Using the definition of Catalan numbers with the gamma function the first and last for example give the identities ... . The last is also equivalent to, and is related to the fact that, which is a consequence of Stirling's approximation. See also