[seqfan] Some conjectured relations in the database (involving Ramanujan's functions)

[Thomas Baruchel]

Dear fellow seqfans,

a new attempt in detecting relations between sequences from the database
by using computing means resulted this time in a list of linearly dependant
sequences. In order to focus on a set of ~1000 interesting sequences where
such relations could be found, I have been working on those having the name
of Ramanujan in their main title (in the downloadable file names.gz).

I didn't spend time looking deeper in the full articles from the database
in order to see whether the following identities/conjectures were already
reported or not; please feel free to add them if you think it may have some
interest. On my own side, I will rather tune my algorithms with new settings
and explore some new ideas. If by chance the following identities bring
something new, I can yield more of them; feel free to ask also.

The most elaborated conjecture in my list involves four sequences :

   -3*A138521(n) + 15*A138520(n) - 16*A213265(n) + 4*A261321(n) = 0

In order to have a normalized way of writing all the identities, please consider
the following output for the same one:

  ['A138521', 'A138520', 'A213265', 'A261321'] --> [-3, 15, -16, 4] (506)
A138520 Expansion of 1 - q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.
A138521 Expansion of chi(-q)^5 / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.
A261321 Expansion of (phi(q) / phi(q^3))^2 in powers of q where phi() is a Ramanujan theta function.

Thus you can read:

   * the list of involved sequences;
   * the vector detected by my program;
   * the norm of the vector (between parentheses);
   * the name of all involved sequences.


All other conjectures involve three sequences:

  ['A233693', 'A132975', 'A128143'] --> [2, -3, 1] (14)
A128143 Expansion of q* (psi(q^9)/phi(q^9))/ (psi(q)/phi(q)) in powers of q where psi(),phi() are Ramanujan theta functions.
A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.
A233693 Expansion of q * psi(-q) * chi(-q^6) * psi(-q^9) / (phi(-q) * phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

  ['A244544', 'A244543', 'A244540'] --> [-2, 1, 1] (6)
A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
A244543 Expansion of phi(q^2) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.

  ['A112128', 'A208604', 'A232358'] --> [2, -1, -1] (6)
A112128 Expansion of phi(q^4) / phi(q) in powers of q where phi() is a Ramanujan theta function.
A208604 Expansion of phi(-q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.
A232358 Expansion of phi(q^2)^2 / (phi(q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function.

  ['A260301', 'A257656', 'A266684'] --> [1, -2, 1] (6)
A257656 Expansion of f(x) * f(x^3) * f(-x^4)^2 * chi(-x^6)^2 in powers of x where chi(), f() are Ramanujan theta functions.
A260301 Expansion of f(-x^3)^3 * psi(x)^3 / psi(x^3)^2 in powers of x where phi(), f() are Ramanujan theta functions.
A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where psi(), f() are Ramanujan theta functions.

  ['A215346', 'A233458', 'A131124'] --> [2, -1, -1] (6)
A131124 Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
A215346 Expansion of (1/q) * phi(-q) * phi(q^4) / (phi(q) * psi(q^8)) in powers of q where phi(), psi() are Ramanujan theta functions.
A233458 Expansion of q^(-1) * (phi(q^2) * phi(-q) / psi(-q^2)^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.

  ['A258034', 'A256280', 'A258256'] --> [3, 1, -4] (26)
A256280 Expansion of phi(q^3)^4 / (phi(q) * phi(q^9)) in powers of q where phi() is a Ramanujan theta function.
A258034 Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function.
A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

  ['A246926', 'A213056', 'A259884'] --> [1, 1, -2] (6)
A213056 Expansion of chi(x) * f(x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.
A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
A259884 Expansion of phi(x) * f(-x^3)^3 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

  ['A257657', 'A260412', 'A132969'] --> [-1, 2, -1] (6)
A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.
A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.
A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x where psi(), f(,) are Ramanujan theta functions.

  ['A247223', 'A133985', 'A121373'] --> [2, -1, -1] (6)
A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.
A133985 Expansion of f(-x, x^2) in powers of x where f(, ) is Ramanujan's general theta function.
A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan theta function.

  ['A258322', 'A256014', 'A256282'] --> [3, 1, -4] (26)
A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.
A256282 Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.
A258322 Expansion of phi(-q) * phi(-q^9) in powers of q where phi() is a Ramanujan theta function.

  ['A260412', 'A257657', 'A260221'] --> [-2, -1, 3] (14)
A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.
A260221 Expansion of phi(x^3)^2 / f(x) in powers of x where phi(), f() are Ramanujan theta functions.
A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x where psi(), f(,) are Ramanujan theta functions.

  ['A261326', 'A213265', 'A233672'] --> [1, 2, -3] (14)
A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.
A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
A261326 Expansion of f(-x^2, -x^4)^2 / (f(x^3, -x^6) * f(-x, x^2)) in powers of x where f(,) is Ramanujan's general theta function.

  ['A115978', 'A253626', 'A253623'] --> [1, -2, 1] (6)
A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.
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.

  ['A131124', 'A233458', 'A215346'] --> [1, 1, -2] (6)
A131124 Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
A215346 Expansion of (1/q) * phi(-q) * phi(q^4) / (phi(q) * psi(q^8)) in powers of q where phi(), psi() are Ramanujan theta functions.
A233458 Expansion of q^(-1) * (phi(q^2) * phi(-q) / psi(-q^2)^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.

  ['A138949', 'A122857', 'A125061'] --> [1, 3, -4] (26)
A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.
A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.
A138949 Expansion of (3 * phi(q^3)^2 - phi(q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

  ['A121455', 'A133691', 'A109506'] --> [-1, -1, 2] (6)
A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.
A121455 Expansion of q*(phi(-q)psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

  ['A233672', 'A213265', 'A261326'] --> [3, -2, -1] (14)
A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.
A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
A261326 Expansion of f(-x^2, -x^4)^2 / (f(x^3, -x^6) * f(-x, x^2)) in powers of x where f(,) is Ramanujan's general theta function.

  ['A261154', 'A132972', 'A128144'] --> [2, -1, -1] (6)
A128144 Expansion of chi(-q)* chi(-q^2)* chi(-q^9)/( chi(-q^3)* chi(q^9)) in powers of q where chi() is a Ramanujan theta function.
A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function.
A261154 Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

  ['A133985', 'A247223', 'A121373'] --> [-1, 2, -1] (6)
A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.
A133985 Expansion of f(-x, x^2) in powers of x where f(, ) is Ramanujan's general theta function.
A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan theta function.


Best regards,

-- 
th. baruchel