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A113652 Expansion of (1 - theta_4(q)^2) / 4 in powers of q.

%I A113652 #18 Jul 14 2024 17:58:50
%S A113652 1,-1,0,-1,2,0,0,-1,1,-2,0,0,2,0,0,-1,2,-1,0,-2,0,0,0,0,3,-2,0,0,2,0,
%T A113652 0,-1,0,-2,0,-1,2,0,0,-2,2,0,0,0,2,0,0,0,1,-3,0,-2,2,0,0,0,0,-2,0,0,2,
%U A113652 0,0,-1,4,0,0,-2,0,0,0,-1,2,-2,0,0,0,0,0,-2,1,-2,0,0,4,0,0,0,2,-2,0,0,0,0,0,0,2,-1,0,-3,2,0,0,-2,0
%N A113652 Expansion of (1 - theta_4(q)^2) / 4 in powers of q.
%D A113652 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(v).
%D A113652 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
%D A113652 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
%D A113652 P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28, Article 269.
%H A113652 G. C. Greubel, <a href="/A113652/b113652.txt">Table of n, a(n) for n = 1..1000</a>
%H A113652 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A113652 a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 4), (1 + (-1)^e)/2 if p == 3 (mod 4).
%F A113652 Expansion of (1 - eta(q)^4 / eta(q^2)^2) / 4 in powers of q.
%F A113652 Moebius transform is period 8 sequence [ 1, -2, -1, 0, 1, 2, -1, 0, ...].
%F A113652 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2 - 2*u3 + u6 - u1^2 + 3*u3^2 + 2*u1*u3 - 4*u2*u6.
%F A113652 G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 + x^k).
%F A113652 G.f.: Sum_{k>0} -(-1)^k * x^k / (1 + x^(2*k)).
%F A113652 G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 + x^(2*k - 1)).
%F A113652 a(n) = -(-1)^n * A002654(n). a(n) = - A104794(n) / 4 unless n = 0.
%F A113652 a(2*n) = - A002654(n). a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 0. a(6*n + 2) = - A122856(n). a(6*n + 4 ) = - A122856(n). - _Michael Somos_, Jun 06 2015
%F A113652 a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(9*n + 3) = a(9*n + 6) = 0. - _Michael Somos_, Jun 06 2015
%e A113652 G.f. = x - x^2 - x^4 + 2*x^5 - x^8 + x^9 - 2*x^10 + 2*x^13 - x^16 + 2*x^17 + ...
%t A113652 a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* _Michael Somos_, Jun 06 2015 *)
%t A113652 a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[4, 0, q]^2) / 4, {q, 0, n}]; (* _Michael Somos_, Jun 06 2015 *)
%t A113652 a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[(1 - EllipticK[m] / (Pi/2)) / 4, {q, 0, n}]]; (* _Michael Somos_, Jun 06 2015 *)
%o A113652 (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker( -4, d)))};
%o A113652 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==1, e+1, !(e%2))))};
%o A113652 (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, if( p==2, 1 - X/(1 - X), 1 / ((1 - X) * (1 - kronecker( -4, p)*X))) )[n])};
%o A113652 (PARI) {a(n) = my(A); if( n<1, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^4 / eta(x^2 + A)^2) / 4, n))};
%Y A113652 Cf. A002654, A053692, A008441, A104794, A113407, A122856, A122865, A258277, A258278.
%K A113652 sign,mult
%O A113652 1,5
%A A113652 _Michael Somos_, Nov 03 2005