cp's OEIS Frontend

A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

%I A133657 #19 Feb 16 2025 08:33:06
%S A133657 1,4,4,0,6,16,8,0,13,24,12,0,14,32,24,0,18,52,20,0,32,48,24,0,31,56,
%T A133657 40,0,30,96,32,0,48,72,48,0,38,80,56,0,42,128,44,0,78,96,48,0,57,124,
%U A133657 72,0,54,160,72,0,80,120,60,0,62,128,104,0,84,192,68,0,96
%N A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
%C A133657 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A133657 Amiram Eldar, <a href="/A133657/b133657.txt">Table of n, a(n) for n = 1..10000</a>
%H A133657 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A133657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A133657 Expansion of (eta(q^2)^5 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
%F A133657 Euler transform of period 8 sequence [ 4, -6, 4, 0, 4, -6, 4, -4, ...].
%F A133657 a(n) is multiplicative with a(2) = 4, a(2^e) = 0 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
%F A133657 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133690.
%F A133657 a(4*n) = 0. a(4*n+2) = 4 * sigma(2*n+1). a(2*n+1) = sigma(2*n+1).
%F A133657 a(n) = -(-1)^n * A121455(n). Convolution square of A113411.
%F A133657 a(2*n + 1) = A008438. a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).
%F A133657 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/16 = 0.6168502... (A222068). - _Amiram Eldar_, Nov 12 2022
%e A133657 G.f. = q + 4*q^2 + 4*q^3 + 6*q^5 + 16*q^6 + 8*q^7 + 13*q^9 + 24*q^10 + ...
%t A133657 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]/2)^2, {q, 0, n}]; (* _Michael Somos_, Oct 30 2015 *)
%t A133657 a[n_] := Switch[IntegerExponent[n, 2], 0, DivisorSigma[1, n], 1, 4*DivisorSigma[1, n/2], _, 0]; Array[a, 100] (* _Amiram Eldar_, Nov 12 2022 *)
%o A133657 (PARI) {a(n) = if( n<1, 0, if( n%2, sigma(n), if( n%4, 4 * sigma(n/2), 0)))};
%o A133657 (PARI) {a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3) )^2, n))};
%Y A133657 Cf. A008438, A097723, A112610, A121455, A133690, A222068.
%K A133657 nonn,mult
%O A133657 1,2
%A A133657 _Michael Somos_, Sep 20 2007