A132320 - OEIS

%I #15 Mar 12 2021 22:24:44

%S 1,-2,1,-2,4,-4,5,-6,9,-12,13,-18,25,-28,33,-44,54,-64,74,-92,114,

%T -132,155,-186,224,-260,303,-360,424,-488,565,-662,770,-888,1018,

%U -1180,1366,-1560,1780,-2048,2345,-2668,3034,-3460,3946,-4468,5052,-5734,6502,-7328

%N McKay-Thompson series of class 22B for the Monster group with a(0) = -2.

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

%H Vaclav Kotesovec, <a href="/A132320/b132320.txt">Table of n, a(n) for n = -1..5000</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 q^-1 * (chi(-q) * chi(-q^11))^2 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q) * eta(q^11) / (eta(q^2) * eta(q^22)))^2 in powers of q.

%F Euler transform of period 22 sequence [ -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (22 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123631.

%F G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(11*k)))^-2.

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

%F A = A058568(n) unless n = 0. Convolution inverse is A123631.

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/11)) / (2*11^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017

%e G.f. = 1/q - 2 + q - 2*q^2 + 4*q^3 - 4*q^4 + 5*q^5 - 6*q^6 + 9*q^7 - ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q, q^2] QPochhammer[ q^11, q^22])^2 / q, {q, 0, n}]; (* _Michael Somos_, Aug 27 2014 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A) / (eta(x^2 + A) * eta(x^22 + A)))^2, n))};

%Y Cf. A058568, A123631.

%K sign

%O -1,2

%A _Michael Somos_, Aug 18 2007