cp's OEIS Frontend

A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A244554 #20 Jun 08 2025 02:31:05
%S A244554 1,1,-2,1,4,-2,0,1,-1,4,-2,-2,4,0,0,1,2,-1,-2,4,0,-2,0,-2,5,4,-4,0,4,
%T A244554 0,0,1,-4,2,0,-1,4,-2,0,4,2,0,-2,-2,4,0,0,-2,1,5,-4,4,4,-4,0,0,-4,4,
%U A244554 -2,0,4,0,0,1,8,-4,-2,2,0,0,0,-1,2,4,-2,-2,0,0
%N A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A244554 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A244554 G. C. Greubel, <a href="/A244554/b244554.txt">Table of n, a(n) for n = 1..2500</a>
%H A244554 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A244554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A244554 Expansion of q * f(-q, -q^7)^2 * phi(q) / psi(-q) = q * f(-q, -q^7)^2 * chi(q)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
%F A244554 Euler transform of period 8 sequence [1, -3, 3, 0, 3, -3, 1, -2, ...].
%F A244554 Moebius transform is period 8 sequence [1, 0, -3, 0, 3, 0, -1, 0, ...].
%F A244554 Convolution product of A244560 and A107635. Convolution product of A000122 and A143259.
%F A244554 a(n) = (A004018(n) - A033715(n)) / 2 = A243747(2*n).
%F A244554 a(2*n) = a(n). a(8*n + 3) = -2 * A033761(n). a(8*n + 5) = 4 * A053692(n). a(8*n + 7) = 0.
%F A244554 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 - 1/sqrt(2))/2 = 0.460075... . - _Amiram Eldar_, Jun 08 2025
%e A244554 G.f. = q + q^2 - 2*q^3 + q^4 + 4*q^5 - 2*q^6 + q^8 - q^9 + 4*q^10 - 2*q^11 + ...
%t A244554 a[ n_] := If[ n < 1, 0, Sum[ {1, 0, -3, 0, 3, 0, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
%t A244554 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];
%o A244554 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 0, -3, 0, 3, 0, -1][d%8 + 1]))};
%o A244554 (PARI) {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * (A - subst(A, x, x^2)) / 2, n))};
%o A244554 (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] + A[2];
%o A244554 (Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] + A[3];
%Y A244554 Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A107635, A121373, A143259, A243747, A244560.
%K A244554 sign
%O A244554 1,3
%A A244554 _Michael Somos_, Jun 30 2014