A138501 Expansion of (eta(q)^2 * eta(q^4)^4 / eta(q^2)^3)^2 in powers of q.

1, -4, 8, -16, 26, -32, 48, -64, 73, -104, 120, -128, 170, -192, 208, -256, 290, -292, 360, -416, 384, -480, 528, -512, 651, -680, 656, -768, 842, -832, 960, -1024, 960, -1160, 1248, -1168, 1370, -1440, 1360, -1664, 1682, -1536, 1848, -1920, 1898, -2112, 2208, -2048, 2353, -2604

Offset

1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 4 sequence [ -4, 2, -4, -6, ...].

a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 3 (mod 4).

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138502.

G.f.: x * (Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4)^2.

a(n) = -(-1)^n * A050470(n).

Example

G.f. = q - 4*q^2 + 8*q^3 - 16*q^4 + 26*q^5 - 32*q^6 + 48*q^7 - 64*q^8 + 73*q^9 + ...

Mathematica

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, #^2 Mod[n/#, 2] (-1)^Quotient[n/#, 2] &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^4]^4 / QPochhammer[ q^2]^3)^2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, d^2 * (n / d % 2) * (-1)^(n / d \ 2)))};
(PARI) {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -4^e, f = (-1)^(p\2); ((p^2)^(e+1) - f^(e+1)) / (p^2 - f))))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n) ; polcoeff( (eta(x + A)^2 * eta(x^4 + A)^4 / eta(x^2 + A)^3)^2, n))};

Crossrefs

Cf. A050470, A138502.

Sequence in context: A013934 A167189 A050470 * A096296 A331243 A180861

Adjacent sequences: A138498 A138499 A138500 * A138502 A138503 A138504

Keyword

sign,mult

Author

Michael Somos, Mar 20 2008

Status

approved