A133098 - OEIS

%I #16 Mar 12 2021 22:24:44

%S 1,-2,1,-1,2,-2,2,-3,5,-5,5,-7,9,-10,11,-14,18,-20,22,-27,32,-36,40,

%T -48,57,-63,70,-82,95,-106,119,-137,158,-175,195,-222,252,-280,311,

%U -352,397,-439,486,-546,611,-676,747,-834,929,-1024,1128,-1253,1389,-1528

%N Expansion of q^(-1) * chi(-q)^2 * chi(-q^15)^2 / (chi(-q^3) * chi(-q^5)) in powers of q where chi() is a Ramanujan theta function.

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

%C McKay-Thompson series of class 30G for the Monster group with a(n) = -2. - _Michael Somos_, Oct 31 2015

%H G. C. Greubel, <a href="/A133098/b133098.txt">Table of n, a(n) for n = -1..1000</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)^2 * eta(q^6) * eta(q^10) * eta(q^15)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^5) * eta(q^30)^2) in powers of q.

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

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123630.

%F G.f.: (1/x) * Product_{k>0} (1 + x^(3*k)) * (1 + x^(5*k)) / ((1 + x^k)^2 * (1 + x^(15*k))^2 ).

%F a(n) = A058618(n) unless n=0. Convolution inverse of A123630.

%F a(n) = -(-1)^n * A145788(n). - _Michael Somos_, Oct 31 2015

%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%F a(2*n) = -A094023(n) = -A123630(n) if n>0. - _Michael Somos_, Oct 22 2017

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

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q^3, q^3] QPochhammer[ -q^5, q^5]) / (QPochhammer[ -q, q] QPochhammer[ -q^15, q^15])^2, {q, 0, n}]; (* _Michael Somos_, Oct 22 2017 *)

%t QP = QPochhammer; a[n_] := SeriesCoefficient[(1/q)* (QP[q]^2*QP[q^6]* QP[q^10]*QP[q^15]^2)/(QP[q^2]^2*QP[q^3]*QP[q^5]*QP[q^30]^2), {q, 0, n}]; (* _G. C. Greubel_, Oct 21 2017 *)

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

%Y Cf. A058618, A094023, A123630, A145788.

%K sign

%O -1,2

%A _Michael Somos_, Sep 10 2007