cp's OEIS Frontend

A225923 Expansion of q^(-1/2) * k(q) * (1 - k(q)^4) * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions.

%I A225923 #35 Jun 29 2025 21:32:32
%S A225923 1,20,-74,-24,157,124,478,-1480,-1198,3044,-480,184,2351,-1720,-3282,
%T A225923 -5728,2480,1776,10326,9560,-8886,-9188,-11618,23664,-16231,-23960,
%U A225923 11686,-9176,60880,16876,-18482,-3768,-35372,-15532,3680,-31960,-4886,47020,-2976,44560
%N A225923 Expansion of q^(-1/2) * k(q) * (1 - k(q)^4) * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions.
%C A225923 In Glaisher (1907) denoted by gamma(m) defined in section 63 on page 38.
%C A225923 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A225923 In Chan and Combes (2024) page 2 is g(z) = eta(z)^8 eta(4z)^4 + 8 eta(4z)^12 identified as the unique newform of weight 6 and level 8 with LMFDB label 8.6.a.a. - _Michael Somos_, Jun 25 2025
%D A225923 J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 38).
%H A225923 G. C. Greubel, <a href="/A225923/b225923.txt">Table of n, a(n) for n = 0..1000</a>
%H A225923 Elisabeth (Yin Ting) Chan and Lewis Combes, <a href="https://arxiv.org/abs/2407.05748">Expressions for Weight 2 Cusp Forms in Holomorphic Eta Quotients</a>, arXiv preprint arXiv:2407.05748 [math.NT], 2024.
%H A225923 LMFDB, <a href="https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/8/6/a/a/">Newform 8.6.a.a</a>.
%H A225923 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A225923 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A225923 Expansion of (psi(x) * phi(-x^2)^2)^4 + 16 * x * (psi(x) * psi(-x)^2)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%F A225923 Expansion of (phi(x)^8 - 256 * x^2 * psi(x^2)^8) * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Jul 20 2013
%F A225923 Expansion of q^(-1/2) * (eta(q)^12 + 32 * q * eta(q)^4 * eta(q^4)^8) in powers of q.
%F A225923 Expansion of q^(-1) * eta(q^4)^4 * (eta(q)^8 + 8 * eta(q^4)^8) in power of q^2. - _Michael Somos_, Jun 25 2025
%F A225923 G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 512 * (t/i)^6 * f(t) where q = exp(2*Pi*i*t).
%F A225923 a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) if p > 2.
%F A225923 G.f.: Product_{k>0} (1 - x^k)^12 + 32 * x * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2)^4.
%F A225923 |a(n)| = A002292(n). a(n) = A000735(n) + 32 * A225872(n).
%e A225923 G.f. = 1 + 20*x - 74*x^2 - 24*x^3 + 157*x^4 + 124*x^5 + 478*x^6 - 1480*x^7 + ...
%e A225923 G.f. = q + 20*q^3 - 74*q^5 - 24*q^7 + 157*q^9 + 124*q^11 + 478*q^13 - 1480*q^15 + ...
%t A225923 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12 + 32 q (QPochhammer[ q] QPochhammer[ q^4]^2)^4, {q, 0, n}];
%t A225923 a[ n_] := SeriesCoefficient[ 8 q QPochhammer[ q^4]^12 + (QPochhammer[ q]^2 QPochhammer[ q^4])^4, {q, 0, 2 n}];
%o A225923 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^12 + 32 * x * eta(x + A)^4 * eta(x^4 + A)^8, n))};
%o A225923 (PARI) {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8 * x * eta(x^4 + A)^12 + eta(x + A)^8 * eta(x^4 + A)^4, n))};
%o A225923 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2))^4 + 16 * x * (eta(x + A) * eta(x^4 + A)^2)^4, n))};
%o A225923 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2  + A)^12 / (eta(x + A)^5 * eta(x^4 + A)^4))^4 - x^2 * (4 * eta(x^4 + A)^4 / eta(x + A))^4, n))}; /* _Michael Somos_, Jul 20 2013 */
%Y A225923 Cf. A000735, A002292, A225872.
%K A225923 sign
%O A225923 0,2
%A A225923 _Michael Somos_, May 20 2013