A110712 -id:A110712 - cp's OEIS frontend

A134080 Expansion of (f(-q^5)^5 / f(-q) + f(q^5)^5 / f(q)) / 2 in powers of q^2 where f() is a Ramanujan theta function.

1, 2, 5, 6, 7, 12, 12, 10, 16, 20, 12, 22, 25, 20, 30, 32, 24, 30, 36, 24, 42, 42, 35, 46, 43, 32, 52, 60, 40, 60, 62, 42, 60, 66, 44, 72, 72, 50, 72, 80, 61, 82, 80, 60, 90, 72, 64, 100, 96, 84, 102, 102, 60, 106, 110, 72, 112, 110, 84, 96, 133, 84, 125, 126

Offset: 0

Michael Somos, Oct 07 2007

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

			G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 7*x^4 + 12*x^5 + 12*x^6 + 10*x^7 + 16*x^8 + ...
G.f. = q + 2*q^3 + 5*q^5 + 6*q^7 + 7*q^9 + 12*q^11 + 12*q^13 + 10*q^15 + 16*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A053723, A110712, A129303, A138483, A138512, A138557, A328717.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ m/d KroneckerSymbol[ 5, d], {d, Divisors @ m}]]]; (* Michael Somos, Jun 14 2014 *)

{a(n) = if( n<0, 0, n = 2*n + 1 ; sumdiv(n, d, kronecker( 5, d) * n / d)) };

Expansion of ( phi(x^5) * psi(x^2) + x * phi(x) * psi(x^10) ) * f(-x^5) * phi(-x^5) / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = (p^(e+1)  + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).
a(n) = A053723(2*n) = A110712(2*n + 1) = A129303(2*n + 1) = A138483(2*n + 1) = A138512(2*n + 1) = A138557(2*n + 1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (5/2) * A328717 = 2*Pi^2/(5*sqrt(5)) = 1.7655285081... . - Amiram Eldar, Nov 23 2023

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.

Original entry on oeis.org

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

Michael Somos, Mar 20 2008

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

			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 + ...

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.

Cf. A110712, A113185.

Cf. A000122, A000700, A010054, A121373.

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 *)

{a(n) = if( n<1, 0, sumdiv(n, d, n/d * kronecker(25, d) * (-1)^((d+2) \ 5) * if(d%4, 1, 5)))};

{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) )))};

{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))};

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

cp's OEIS Frontend

A134080 Expansion of (f(-q^5)^5 / f(-q) + f(q^5)^5 / f(q)) / 2 in powers of q^2 where f() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula