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 A344014 Coefficients (negated) of expansion of the operator U_2 applied to j^-1, the inverse of the Klein j-invariant, with respect to powers of j^-1. 4
 744, 140914688, 16324041375744, 1528926232501026816, 127072326069001429975040, 9781118992341002031206498304, 713663408582010002941475567960064, 50057559997415568004332170039751868416, 3406371342315881760006472823773108302249984 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It is not clear if all the coefficients are integers. LINKS Robin Visser, Table of n, a(n) for n = 1..200 Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356. EXAMPLE From Robin Visser, Jul 29 2023: (Start) A q-expansion for the inverse of Klein's j-invariant is given by: j^-1 = q - 744*q^2 + 356652*q^3 - 140361152*q^4 + 49336682190*q^5 - 16114625669088*q^6 + O(q^7). Thus a q-expansion for U_2 operated on j^-1 is: U_2 j^-1 = -744*q - 140361152*q^2 - 16114625669088*q^3 + O(q^4). Computing q-expansions for j^-2 and j^-3 gives j^-2 = q^2 - 1488*q^3 + O(q^4), and j^-3 = q^3 + O(q^4). This yields an expansion for U_2 j^_1 in terms of powers of j^-1 as U_2 j^-1 = -744*j^-1 - 140914688*j^-2 - 16324041375744*j^-3 - ..., which gives the first three terms as a(1) = 744, a(2) = 140914688, and a(3) = 16324041375744. (End) PROG (Sage) def a(n): j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 2*n)]) j2 = product([x]+[(1-x^k)^24 for k in range(1, 2*n)]) jinv = (j2/j1^3).taylor(x, 0, 2*n) U2jinv = sum([jinv.coefficient(x^(2*k))*x^k for k in range(0, 2*n)]) for k in range(1, n): c = U2jinv.taylor(x, 0, k).coefficient(x^k) U2jinv -= c*(jinv^k) return -U2jinv.taylor(x, 0, n).coefficient(x^n) # Robin Visser, Jul 29 2023 CROSSREFS Cf. A000521, A066395, A344015, A344016, A344017. Sequence in context: A091406 A066396 A099819 * A051978 A235732 A235513 Adjacent sequences: A344011 A344012 A344013 * A344015 A344016 A344017 KEYWORD nonn AUTHOR N. J. A. Sloane, Jun 17 2021 EXTENSIONS More terms from Robin Visser, Jul 29 2023 STATUS approved

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Last modified August 3 10:11 EDT 2024. Contains 374885 sequences. (Running on oeis4.)