A104794 Expansion of theta_4(q)^2 in powers of q.

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).

In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.

References

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.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of (1-k^2)^(1/2) K(k^2) / (Pi/2) in powers of q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind.

Expansion of K(k^2) / (Pi/2) in powers of -q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind. - Michael Somos, Jun 08 2015

Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.

Euler transform of period 2 sequence [ -4, -2, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.

Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008441.

G.f.: theta_4(q)^2 = (Sum_{k in Z} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.

G.f.: 1 + 4 * Sum_{k>0} (-x)^k / (1 + x^(2*k)). - Michael Somos, Jun 08 2015

a(4*n + 3) = 0. a(n) = (-1)^n * A004018(n) = a(2*n). a(4*n + 1) = -4 * A008441(n). a(n) = -4 * A113652(n) unless n=0. a(6*n + 2) = 4 * A122865(n)). a(6*n + 4) = 4 * A122856(n). a(8*n + 1) = -4 * A113407(n). a(8*n + 5) = -8 * A053692(n).

a(n) = a(9*n) = A204531(8*n) = A246950(8*n) = A256014(9*n) = A258210(n). - Michael Somos, Jun 08 2015

Convolution inverse of A001934. Convolution with A000729 is A227695. - Michael Somos, Jun 08 2015

G.f.: 2 * Sum_{k in Z} (-1)^k * x^(k*(k + 1)/2) / (1 + x^k). - Michael Somos, Nov 05 2015

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017

G.f.: exp(2*Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

Example

G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 4 DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, (-1)^n * 4 * sumdiv(n, d, (d%4==1) - (d%4==3)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 / eta(x^2 + A)^2, n ))};
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))};
(Magma) A := Basis( ModularForms( Gamma1(8), 1), 100); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Jan 31 2015 */
(Julia) # JacobiTheta4 is defined in A002448.
A104794List(len) = JacobiTheta4(len, 2)
A104794List(102) |> println # Peter Luschny, Mar 12 2018

Crossrefs

Cf. A000203, A000729, A001934, A002131, A004018, A008441, A053692, A113407, A113652, A122856, A122865, A204531, A227695, A246950, A256014, A258210.

Sequence in context: A245971 A279365 A164613 * A004018 A253185 A028658

Adjacent sequences: A104791 A104792 A104793 * A104795 A104796 A104797

Keyword

sign

Author

Michael Somos, Mar 26 2005

Status

approved