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A132321 McKay-Thompson series of class 30C for the Monster group with a(0) = -1. 4

%I #20 Mar 12 2021 22:24:44

%S 1,-1,0,-2,2,-2,3,-2,5,-6,5,-6,9,-10,10,-16,17,-18,25,-26,31,-38,37,

%T -48,60,-62,68,-84,95,-104,125,-134,154,-182,192,-220,257,-274,309,

%U -360,394,-434,492,-544,607,-688,740,-824,944,-1018,1123,-1266,1377,-1524

%N McKay-Thompson series of class 30C for the Monster group with a(0) = -1.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H Seiichi Manyama, <a href="/A132321/b132321.txt">Table of n, a(n) for n = -1..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of eta(q) * eta(q^3) * eta(q^5) * eta(q^15) / (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) in powers of q.

%F Euler transform of period 30 sequence [ -1, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1, 0, -1, 0, -4, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 4 * u + 2 * u * v.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123632.

%F G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 + x^(5*k)) * (1 + x^(15*k)))^-1.

%F Expansion of q^-1 * chi(-q) * chi(-q^3) * chi(-q^5) * chi(-q^15) in powers of q where chi() is a Ramanujan theta function.

%F a(n) = A058614(n) unless n = 0.

%F Convolution inverse of A123632.

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/15)) / (2*15^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018

%e G.f. = 1/q - 1 - 2*q^2 + 2*q^3 - 2*q^4 + 3*q^5 - 2*q^6 + 5*q^7 - 6*q^8 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q^3, q^6] QPochhammer[ q^5, q^10] QPochhammer[ q^15, q^30] / q, {q, 0, n}]; (* _Michael Somos_, Nov 01 2015 *)

%o (PARI) {a(n) = my(A); if(n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^15 + A) / (eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)), n))};

%Y Cf. A058614, A123632.

%K sign

%O -1,4

%A _Michael Somos_, Aug 18 2007

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)