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%I #15 Dec 29 2023 03:50:33
%S 1,8,26,48,73,120,170,208,290,360,384,528,651,656,842,960,960,1248,
%T 1370,1360,1682,1848,1898,2208,2353,2320,2810,3120,2880,3480,3722,
%U 3504,4420,4488,4224,5040,5330,5208,5760,6240,5905,6888,7540,6736,7922,8160,7680
%N Expansion of phi(q)^2*psi(q)^4 in powers of q where phi(),psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H G. C. Greubel, <a href="/A122854/b122854.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of q^(-1/2)eta(q^2)^18/(eta(q)^8*eta(q^4)^4) in powers of q.
%F Euler transform of period 4 sequence [ 8, -10, 8, -6, ...].
%F a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4), b(p^e) = ((p^2)^(e+1)-(-1)^(e+1))/(p^2+1) if p == 3 (mod 4).
%F G.f.: Sum_{k>0 odd} k^2*x^k/(1+x^(2k)) = Product_{k>0} (1-x^(2k))^6*(1+x^k)^8/(1+x^(2k))^4.
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^3/24 = 1.291928... (A152584). - _Amiram Eldar_, Dec 29 2023
%t a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(1/2)]^4 * EllipticTheta[3, 0, q]^2)/16, {q, 0, n}]; Table[A122854[n], {n, 0, 50}] (* _G. C. Greubel_, Jan 04 2018 *)
%o (PARI) {a(n)= local(A, p, e, f); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, f=(-1)^(p\2); (p^(2*e+2)-f^(e+1))/(p^2-f)))))}
%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^18/(eta(x+A)^8*eta(x^4+A)^4), n))}
%Y A050458(2n+1) = A050470(2n+1) = a(n).
%Y Cf. A152584.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Sep 14 2006
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