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Metadata 
bibkey: hanna2023bilateral
authors: Paul D. Hanna
year: 2023
title: "OEIS A363558, A363559, A363561 and A363569: bilateral power sums"
doi: null
url: https://oeis.org/A363558
claim: "Eight conjectures on the coefficients of Sum_{n=-oo..+oo} x^n * (c + x^n)^(2*n), at powers of two and primes greater than three, for c = 2, 3, sqrt(2), i."
strata_touched:
  - D5/S1/Recurrence/HannaBilateral/OeisA363558PowerOfTwo
  - D5/S1/Recurrence/HannaBilateral/OeisA363558Prime
  - D5/S1/Recurrence/HannaBilateral/OeisA363559PowerOfTwo
  - D5/S1/Recurrence/HannaBilateral/OeisA363559Prime
  - D5/S1/Recurrence/HannaBilateral/OeisA363561PowerOfTwo
  - D5/S1/Recurrence/HannaBilateral/OeisA363561Prime
  - D5/S1/Recurrence/HannaBilateral/OeisA363569PowerOfTwo
  - D5/S1/Recurrence/HannaBilateral/OeisA363569Prime
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A363558

Related source entries: https://oeis.org/A363559, https://oeis.org/A363561, https://oeis.org/A363569. These are unpaginated OEIS entries; the statements below occur in their NAME and FORMULA fields.

Source statements

  • A363558, NAME: “Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).”
  • A363558, FORMULAS: “a(2^n) = 4^(2^n) for n > 0 (conjecture).” “a(p) = p*2^(p-1) + 4^p for primes p > 3 (conjecture).”
  • A363559, NAME: “Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).”
  • A363559, FORMULAS: “a(2^n) = 9^(2^n) for n > 0 (conjecture).” “a(p) = p*3^(p-1) + 9^p for primes p > 3 (conjecture).”
  • A363561, NAME: “G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).”
  • A363561, FORMULAS: “a(2^n) = 2^(2^n) for n > 0 (conjecture).” “a(p) = p*2^((p-1)/2) + 2^p for primes p > 3 (conjecture).”
  • A363569, NAME: “Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.”
  • A363569, FORMULAS: “a(2^n) = 1 for n > 0 (conjecture).” “a(p) = p*(-1)^((p-1)/2) - 1 for primes p > 3 (conjecture).”

Coefficients and scope

The formal coefficients are complex numbers. The nonnegative integer terms are expanded as polynomials. For a negative index -k, with k positive, the summand is X^(2k^2-k) times the formal inverse of (1+cX^k)^(2*k). At degree m only nonnegative indices at most m and negative magnitudes at most m can contribute. No analytic convergence assertion is needed.

The coefficient formulas at positive dyadic powers hold for every complex c: coeffA(c,2^N)=c^(2^(N+1)) for N>0. For prime p>3, the formula is coeffA(c,p)=pc^(p-1)+c^(2p). The displayed OEIS conjectures specialize these identities to the four parameters above. Completeness of the literature screen is ASSUMED-UNVERIFIED.

Numerical evidence

The independent experiment is in the private repository the-omega-institute/trureturing-experiments at commit 1133ccd5c01a70de0997056d1ec6bdbb3c8b6a32, under docs/reports/oeis-a363558-a363559-a363561-a363569-bilateral-power-sums/. Its check.py has SHA-256 b39e40018fd0826204c5ecc5d01373129e88b32b0b1556b44e8631c2b0431c04. The recorded command python3 check.py 300 12 exits 0 with RESULT PASS. It checks symbolic coefficients through degree 14, dyadic exponents 1 through 12 and primes at most 300; these finite readings are distinct from the universal Lean statements.