login
G.f.: Product_{k>0} (1 - x^k)^4 * (1 - (-x)^k)^8.
2

%I #39 Mar 12 2021 22:24:46

%S 1,4,-10,-56,29,332,30,-1064,-302,1940,288,-1960,1071,1192,-1938,-736,

%T -2000,-1488,5014,7288,4170,-10644,-8482,11184,-12647,-15544,15590,

%U 9992,25424,4604,-26610,2472,-28972,3140,26464,-39416,31338,24764,-25248,-16176

%N G.f.: Product_{k>0} (1 - x^k)^4 * (1 - (-x)^k)^8.

%C This is Glaisher's alpha(m) for odd values of m. - _N. J. A. Sloane_, Nov 24 2018

%H G. C. Greubel, <a href="/A225543/b225543.txt">Table of n, a(n) for n = 0..1000</a>

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 37).

%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>

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>

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

%F Expansion of phi(-x^2)^8 * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. (Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).)

%F Expansion of f(x)^8 * f(-x)^4 in powers of x where f() is a Ramanujan theta function.

%F Expansion of q^(-1/2) * (eta(q^2)^6 / (eta(q) * eta(q^4)^2))^4 in powers of q.

%F Euler transform of period 4 sequence [ 4, -20, 4, -12, ...].

%F |a(n)| = A002290(n).

%e 1 + 4*x - 10*x^2 - 56*x^3 + 29*x^4 + 332*x^5 + 30*x^6 - 1064*x^7 + ...

%e or

%e q + 4*q^3 - 10*q^5 - 56*q^7 + 29*q^9 + 332*q^11 + 30*q^13 - 1064*q^15 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^4 QPochhammer[ -q]^8, {q, 0, n}]

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^4]^2 EllipticTheta[ 2, 0, q] / 2)^4, {q, 0, 1 + 2 n}]

%o (PARI) {a(n) = local(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, n))}

%Y Cf. A002290.

%K sign

%O 0,2

%A _Michael Somos_, May 17 2013

%E Entry revised by _N. J. A. Sloane_, Apr 27 2014