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 A205962 McKay-Thompson series of class 30D for the Monster group with a(0) = 2. 4
 1, 2, 3, 4, 5, 10, 15, 22, 29, 36, 53, 72, 99, 128, 160, 212, 272, 354, 448, 556, 703, 874, 1096, 1356, 1662, 2050, 2501, 3060, 3716, 4492, 5444, 6550, 7882, 9436, 11262, 13460, 16013, 19034, 22536, 26616, 31450, 37048, 43602, 51164, 59905 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = -1..1000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(-1) * ((chi(-q^3) * chi(-q^15)) / (chi(-q) * chi(-q^5)))^2 in powers of q where chi() is a Ramanujan theta function. Expansion of (eta(q^2) * eta(q^3) * eta(q^10) * eta(q^15) / (eta(q) * eta(q^5) * eta(q^6) * eta(q^30)))^2 in powers of q. Euler transform of period 30 sequence [ 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (w^2 - v) - 4*u*v * (v - 1)^2. - Michael Somos, Jun 09 2012 G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = f(t) where q = exp(2 Pi i t). G.f.: (1/x) * (Product_{k>0} (1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*k))))^2. a(n) = A058615(n) unless n=0. Convolution square of A058729. a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015 EXAMPLE 1/q + 2 + 3*q + 4*q^2 + 5*q^3 + 10*q^4 + 15*q^5 + 22*q^6 + 29*q^7 + ... MATHEMATICA nmax = 50; CoefficientList[Series[Product[((1 + x^k) * (1 + x^(5*k)) / ((1 + x^(3*k)) * (1 + x^(15*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^2] *eta[q^3]*eta[q^10]*eta[q^15]/(eta[q]*eta[q^5]*eta[q^6]*eta[q^30]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 06 2018 *) PROG (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^10 + A) * eta(x^15 + A) / (eta(x + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^30 + A)))^2, n))} CROSSREFS Cf. A058615, A058729. Sequence in context: A140730 A273732 A282032 * A134220 A179146 A099161 Adjacent sequences:  A205959 A205960 A205961 * A205963 A205964 A205965 KEYWORD nonn AUTHOR Michael Somos, Feb 02 2012 STATUS approved

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Last modified August 26 03:38 EDT 2019. Contains 326324 sequences. (Running on oeis4.)