A253626 Expansion of psi(q^2) * f(q, q^2)^2 / f(q, q^5) in powers of q where psi(), f() are Ramanujan theta functions.

1, 1, 3, 1, 3, 0, 3, 2, 3, 1, 0, 0, 3, 2, 6, 0, 3, 0, 3, 2, 0, 2, 0, 0, 3, 1, 6, 1, 6, 0, 0, 2, 3, 0, 0, 0, 3, 2, 6, 2, 0, 0, 6, 2, 0, 0, 0, 0, 3, 3, 3, 0, 6, 0, 3, 0, 6, 2, 0, 0, 0, 2, 6, 2, 3, 0, 0, 2, 0, 0, 0, 0, 3, 2, 6, 1, 6, 0, 6, 2, 0, 1, 0, 0, 6, 0, 6

Offset

0,3

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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

Expansion of (a(q) + 3*a(q^2) + 2*a(q^4)) / 6 = b(q^4) * (-b(q) + 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.

Expansion of eta(q^3)^3 * eta(q^4)^3 / ( eta(q) * eta(q^2) * eta(q^6) * eta(q^12) ) in powers of q.

Euler transform of period 12 sequence [ 1, 2, -2, -1, 1, 0, 1, -1, -2, 2, 1, -2, ...].

Moebius transform is period 12 sequence [ 1, 2, 0, 0, -1, 0, 1, 0, 0, -2, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

a(n) is multiplicative with a(0) = 1, a(2^e) = 3 if e > 0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).

G.f.: 1 + Sum_{k>0} (3 - (k mod 2)*2) * (q^k + q^(3*k)) / (1 + q^(2*k) + q^(4*k)).

G.f.: Product_{k>0} (1 - q^(3*k))^3 * (1 - q^(4*k))^3 / ( (1 - q^k) * (1 - q^(2*k)) * (1 - q^(6*k)) * (1 - q^(12*k)) ).

a(n) = (-1)^n * A253625(n). a(2*n) = A107760(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(4*n + 1) = A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = A112605(n).

a(6*n + 1) = A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). a(12*n + 10) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Jan 21 2024

Example

G.f. = 1 + q + 3*q^2 + q^3 + 3*q^4 + 3*q^6 + 2*q^7 + 3*q^8 + q^9 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ (-1)^ Quotient[ d, 3] {1, 1, 0}[[ Mod[d, 3, 1] ]] {1, 2}[[ Mod[n/d, 2, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q] QPochhammer[ -q^3] (QPochhammer[ q^3, q^6] QPochhammer[ -q^2, q^2])^4, {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-12, d) + if(d%2, 0, 2 * kronecker(-12, d/2))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^3 / ( eta(x + A) * eta(x^2 + A) * eta(x^6 + A) * eta(x^12 + A) ), n))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 1, if( p%6 == 1, e+1, 1-e%2))))))};
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 86); A[1] + A[2] + 3*A[3] + A[4] + 3*A[5];

Crossrefs

Cf. A033687, A033762, A093602, A097195, A107760, A112604, A112605, A121361, A122861, A123884, A253625.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A201659 A253625 A244375 * A112298 A011430 A073747

Adjacent sequences: A253623 A253624 A253625 * A253627 A253628 A253629

Keyword

nonn,mult

Author

Michael Somos, Jan 06 2015

Status

approved