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A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function. 3
1, 3, 3, 2, 3, 4, 2, 0, 3, 5, 4, 2, 2, 4, 0, 0, 3, 6, 5, 2, 4, 0, 2, 0, 2, 7, 4, 4, 0, 4, 0, 0, 3, 4, 6, 0, 5, 4, 2, 0, 4, 6, 0, 2, 2, 4, 0, 0, 2, 3, 7, 4, 4, 4, 4, 0, 0, 4, 4, 2, 0, 4, 0, 0, 3, 8, 4, 2, 6, 0, 0, 0, 5, 6, 4, 2, 2, 0, 0, 0, 4, 7, 6, 2, 0, 8, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 = 0..2500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-q^3, -q^5)^2 * phi(q) / psi(-q) = f(-q^3, -q^5)^2 * chi(q)^3 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

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

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

Convolution product of A244526 and A107635. Convolution product of A000122 and A093709.

a(n) = (A004018(n) + A033715(n)) / 2 = A244543(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. = 1 + 3*q + 3*q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 2*q^6 + 3*q^8 + 5*q^9 + ...

MATHEMATICA

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {3, 0, -1, 0, 1, 0, -3, 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, n==0, sumdiv(n, d, [0, 3, 0, -1, 0, 1, 0, -3][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[0] + 3*A[1] + 3*A[2];

(MAGMA) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 3*A[2] + 3*A[3];

CROSSREFS

Cf. A000122, A004018, A033715, A033761, A053692, A093709, A107635, A244526, A244543.

Sequence in context: A245885 A188834 A276575 * A276585 A057853 A240088

Adjacent sequences:  A244537 A244538 A244539 * A244541 A244542 A244543

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 29 2014

STATUS

approved

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Last modified February 24 22:57 EST 2020. Contains 332216 sequences. (Running on oeis4.)