A100130 - OEIS

%I #23 Mar 12 2021 22:24:43

%S 1,-24,300,-2624,18126,-105504,538296,-2471424,10400997,-40674128,

%T 149343012,-519045888,1718732998,-5451292992,16633756008,-49010118656,

%U 139877936370,-387749049720,1046413709980,-2754808758144,7087483527072

%N Expansion of (eta(q) * eta(q^4) / eta(q^2)^2)^24 in powers of q.

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

%H G. C. Greubel, <a href="/A100130/b100130.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 q / chi(q)^24 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of lambda * (1 - lambda) / 16 in powers of q.

%F Euler transform of period 4 sequence [ -24, 24, -24, 0, ...].

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

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 4096 * (u*v)^4 + (u*v)^2 * (1791 + 2352 * (u + v) - 10496 * u*v) - u*v * (1 - 48 * (u + v) + 96 * (u^2 + v^2)) + u^3 + v^3.

%F G.f.: x * (Product_{k>0} (1 + (-x)^k))^24 = x / (Product_{k>0} (1 + x^(2*k - 1)))^24.

%F a(n) = -(-1)^n * A014103(n). Convolution inverse of A097340. Series reversion of A195130.

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(2*n)) / (4096 * 2^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

%F G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 08 2018

%e G.f. = q - 24*q^2 + 300*q^3 - 2624*q^4 + 18126*q^5 - 105504*q^6 + ...

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) m / 16, {q, 0, n}]];

%t a[ n_] := SeriesCoefficient[ q / Product[ 1 + q^k, {k, 1, n, 2}]^24, {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ q / QPochhammer[ -q, q^2]^24, {q, 0, n}];

%o (PARI) {a(n) = polcoeff( x * prod(k=1, n, 1 + (-x)^k, 1 + x * O(x^n))^24, n)};

%o (PARI) {a(n) = my(A); if( n<1, 0,  n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^24, n))};

%Y Cf. A014103, A097340, A195130.

%K sign

%O 1,2

%A _Michael Somos_, Nov 06 2004

%E Swapped a formula with definition to make this clearer. - _N. J. A. Sloane_, Nov 26 2018