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

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, 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, -2, 0, 4, 0, 0, 1, 8, -4, -2, 2, 0, 0, 0, -1, 2, 4, -2, -2, 0, 0

Offset

1,3

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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.

Euler transform of period 8 sequence [1, -3, 3, 0, 3, -3, 1, -2, ...].

Moebius transform is period 8 sequence [1, 0, -3, 0, 3, 0, -1, 0, ...].

Convolution product of A244560 and A107635. Convolution product of A000122 and A143259.

a(n) = (A004018(n) - A033715(n)) / 2 = A243747(2*n).

a(2*n) = a(n). a(8*n + 3) = -2 * A033761(n). a(8*n + 5) = 4 * A053692(n). a(8*n + 7) = 0.

Example

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 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ {1, 0, -3, 0, 3, 0, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 0, -3, 0, 3, 0, -1][d%8 + 1]))};
(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))};
(Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] + A[2];
(Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] + A[3];

Crossrefs

Cf. A000122, A004018, A033715, A033761, A053692, A107635, A143259, A243747, A244560.

Sequence in context: A177413 A009832 A016445 * A364953 A194735 A130544

Adjacent sequences: A244551 A244552 A244553 * A244555 A244556 A244557

Keyword

sign

Author

Michael Somos, Jun 30 2014

Status

approved