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A007263 Coefficients of completely replicable function "6d".
(Formerly M4995)
5
1, 0, 16, -8, 0, 128, 28, 0, 576, -64, 0, 2048, 134, 0, 6304, -288, 0, 17408, 568, 0, 44416, -1024, 0, 106496, 1809, 0, 242480, -3152, 0, 528896, 5316, 0, 1112128, -8704, 0, 2265088, 13990, 0, 4486112 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,3
COMMENTS
Original name was "McKay-Thompson series of class 6d for Monster" but this series is non-monstrous. Refer to table in Alexander, et. al. 1990. - Michael Somos, Nov 20 2019
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 20 2019
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000 (terms -1..99 from G. A. Edgar)
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, in Groups, Combinatorics & Geometry, (Durham, 1990), pp. 87--98, London Math. Soc. Monograph No. 165.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
FORMULA
Expansion of (eta(q^3)/eta(q^6))^8 + 16*(eta(q^6)/eta(q^3))^8 in powers of q. - G. A. Edgar, Mar 10 2017
G.f.: T6d = T6F + 16/T6F with T6F as in A007259. - G. A. Edgar, Mar 10 2017
a(3*n - 1) = A007259(n). a(3*n + 1) = 16*A022573(n). - Michael Somos, Nov 20 2019
EXAMPLE
T6d = 1/q + 16*q - 8*q^2 + 128*q^4 + 28*q^5 + 576*q^7 - 64*q^8 + ...
MATHEMATICA
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[(eta[q^3]/ eta[q^6])^8 + 16*(eta[q^6]/eta[q^3])^8, {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Jan 25 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x^6 + A))^8; polcoeff( A + 16*x^2/A, n))}; /* Michael Somos, Nov 20 2019 */
CROSSREFS
Sequence in context: A070580 A298618 A028579 * A033336 A204325 A355873
KEYWORD
sign
AUTHOR
EXTENSIONS
More terms from G. A. Edgar, Mar 03 2017
Name change from Michael Somos, Nov 20 2019
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)