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 A101127 McKay-Thompson series of class 12D for the Monster group. 3
 1, 8, 28, 64, 134, 288, 568, 1024, 1809, 3152, 5316, 8704, 13990, 22208, 34696, 53248, 80724, 121240, 180068, 264448, 384940, 556064, 796760, 1132544, 1598789, 2243056, 3127360, 4333568, 5971922, 8188096, 11170160, 15163392, 20491033 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). REFERENCES D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 Index entries for McKay-Thompson series for Monster simple group Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Eric Weisstein's World of Mathematics, Infinite Product FORMULA Expansion of q^(1/3) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^8 in powers of q. Euler transform of period 4 sequence [8, -8, 8, 0, ...]. Given g.f. A(x), B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v*(u^3+v^3) -(u*v)^3 + 15*(u*v)^2 - 32*u*v + 16. G.f.: (Product_{k>0} (1 + x^(2*k-1)))^8. A007259(n) = (-1)^n * a(n). Convolution square of A112160. a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015 Expansion of chi(x)^8 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 12 2017 G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018 EXAMPLE T12D = 1/q + 8*q^2 + 28*q^5 + 64*q^8 + 134*q^11 + 288*q^14 + 568*q^17 + ... MATHEMATICA nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^8, {x, 0, n}]; (* Michael Somos, Sep 12 2017 *) PROG (PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^8, n))}; (PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + A)^8, n))}; CROSSREFS cf. A007259, A112160. Sequence in context: A002408 A340964 A353325 * A007259 A134747 A083013 Adjacent sequences: A101124 A101125 A101126 * A101128 A101129 A101130 KEYWORD nonn AUTHOR Michael Somos, Dec 02 2004 STATUS approved

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Last modified June 18 04:26 EDT 2024. Contains 373468 sequences. (Running on oeis4.)