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 A112194 McKay-Thompson series of class 54c for the Monster group. 3
 1, 1, 2, 1, 3, 3, 4, 4, 6, 8, 10, 11, 14, 16, 20, 23, 28, 32, 40, 45, 55, 61, 74, 83, 98, 111, 130, 148, 172, 195, 224, 253, 291, 327, 374, 420, 481, 539, 612, 683, 775, 865, 976, 1087, 1224, 1365, 1530, 1701, 1902, 2113, 2358, 2613, 2910, 3221, 3584, 3960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994). William J. Keith, Partitions into parts simultaneously regular, distinct, and/or flat, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence. FORMULA Expansion of q^(1/6) * eta(q^3)^2 / (eta(q) * eta(q^9)) in powers of q. - Michael Somos, Aug 28 2015 Euler transform of period 9 sequence [ 1, 1, -1, 1, 1, -1, 1, 1, 0, ...]. - Michael Somos, Aug 28 2015 G.f. is a period 1 Fourier Series which satisifies f(-1 / (324 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 28 2015 G.f. is (1/q) f(q^18)^2/(f(q^6) f(q^54)) with f(q) the product of 1-q^k over all k; this is (1/q) times the generating function for partitions with parts not divisible by 3 which further must appear fewer than 3 times, magnified by the substitution q^6 for q. - William J. Keith, Jul 21 2015 a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015 EXAMPLE G.f. = 1 + x + 2*x^2 + x^3 + 3*x^4 * 3*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + ... T54c = 1/q + q^5 + 2*q^11 + q^17 + 3*q^23 + 3*q^29 + 4*q^35 + 4*q^41 +... MATHEMATICA Series[Product[((1-q^(3k))^2)/((1-q^k)(1-q^(9k))), {k, 1, 55}], {q, 0, 55}] (* William J. Keith, Jul 21 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^9]) , {x, 0, n}]; (* Michael Somos, Aug 28 2015 *) PROG (PARI)  N=66; q='q+O('q^N); Vec(prod(n=1, N, if(n%3, 1+q^n+q^(2*n), 1))) /* Joerg Arndt, Jul 22 2015 */ (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A) * eta(x^9 + A)), n))}; /* Michael Somos, Aug 28 2015 */ CROSSREFS Sequence in context: A323054 A339396 A027157 * A238788 A083041 A318611 Adjacent sequences:  A112191 A112192 A112193 * A112195 A112196 A112197 KEYWORD nonn AUTHOR Michael Somos, Aug 28 2005 STATUS approved

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Last modified August 1 13:34 EDT 2021. Contains 346391 sequences. (Running on oeis4.)