A138483 Expansion of (phi(q)^3 * phi(q^5) - phi(q) * phi(q^5)^3) / 4 in powers of q where phi() is a Ramanujan theta function.

1, 3, 2, 1, 5, 6, 6, 7, 7, 15, 12, 2, 12, 18, 10, 9, 16, 21, 20, 5, 12, 36, 22, 14, 25, 36, 20, 6, 30, 30, 32, 23, 24, 48, 30, 7, 36, 60, 24, 35, 42, 36, 42, 12, 35, 66, 46, 18, 43, 75, 32, 12, 52, 60, 60, 42, 40, 90, 60, 10, 62, 96, 42, 41, 60, 72, 66, 16, 44

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, 2019.

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

Formula

Expansion of q * phi(q) * chi(q) * f(q^5) * f(-q^10)^2 in powers of q where phi(), chi(), f() are Ramanujan theta functions.

Expansion of eta(q^2)^7 * eta(q^10)^5 / (eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [3, -4, 3, -1, 4, -4, 3, -1, 3, -8, 3, -1, 3, -4, 4, -1, 3, -4, 3, -4, ...].

a(n) is multiplicative with a(2^e) = (2^(e+1) - 5*(-1)^e) / 3 if e>0, a(5^e) = 5^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 80^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113185.

G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^3 * (1 - x^(5*k))^3 * (1 + x^(10*k-5))^4 * (1 + x^(10*k))^3.

a(n) = -(-1)^n * A110712(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(5*sqrt(5)) = 0.882764... . - Amiram Eldar, Nov 24 2023

Example

G.f. = q + 3*q^2 + 2*q^3 + q^4 + 5*q^5 + 6*q^6 + 6*q^7 + 7*q^8 + 7*q^9 + ...

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 25, #] (-1)^Quotient[# + 2, 5] If[ Mod[#, 4] > 0, 1, 5] &]]; (* Michael Somos, Sep 27 2015 *)
a[ n_] := SeriesCoefficient[ q EllipticTheta[3, 0, q] QPochhammer[ -q, q^2] QPochhammer[ -q^5] QPochhammer[ q^10]^2, {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, q]^3 EllipticTheta[3, 0, q^5] - EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5]^3) / 4, {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, n/d * kronecker(25, d) * (-1)^((d+2) \ 5) * if(d%4, 1, 5)))};
(PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (2^(e+1) - 5*(-1)^e) / 3, f = kronecker(5, p); (p^(e+1) - f^(e+1)) / (p - f) )))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^10 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A110712, A113185.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A209577 A139377 A368607 * A110712 A065366 A360364

Adjacent sequences: A138480 A138481 A138482 * A138484 A138485 A138486

Keyword

nonn,easy,mult

Author

Michael Somos, Mar 20 2008

Status

approved