bibkey: hanna2019a323689
authors: Paul D. Hanna
year: 2019
title: "OEIS A323689: Gaussian generating function and square parity"
doi: null
url: https://oeis.org/A323689
claim: "Odd terms occur only at n = k^2 for k >= 0 (conjecture); a(n^2) = 1 (mod 2) for n >= 0."
strata_touched:
  - D5/S1/Recurrence/Algebraic/HannaGaussianSquareParity
license: citation-only
triage: anchor

OEIS A323689

Verified locator

Source: https://oeis.org/A323689

The unpaginated native text states:

%I A323689 #39 Feb 21 2019 20:51:19
%N A323689 G.f.: Sum_{n>=0} (x^n + i)^n / (1 + i*x^n)^(n+1), in which the constant term is taken to be 1.
%C A323689 It is remarkable that the generating function results in a power series in x with only real coefficients.
%C A323689 Odd terms occur only at n = k^2 for k >= 0 (conjecture);
%F A323689 If the constant term in each of the following sums is taken to be 1, then each can be expanded into a power series in x given by g.f. A(x) = Sum_{n>=0} a(n)*x^n.
%F A323689 (1a) A(x) = Sum_{n>=0} (x^n + i)^n / (1 + i*x^n)^(n+1).
%F A323689 a(n^2) = 1 (mod 2) for n >= 0.
%e A323689 S(N) = Sum_{n=0..N} (x^n + i)^n / (1 + i*x^n)^(n+1) = i^N/(1+i) + 3*x - 14*x^3 + 13*x^4 + 22*x^5 - 30*x^7 - 82*x^8 + 101*x^9 - 46*x^11 + 170*x^12 + ...
%e A323689 here, we ignore the coefficient of x^0 and set a(0) = 1.
%o A323689 (PARI) {a(n) = my(SUM = sum(m=0, n, (x^m + I +x*O(x^n))^m / (1 + I*x^m +x*O(x^n))^(m+1) ) ); polcoeff(1 + SUM - I^n/(1+I), n)}
%O A323689 0,2
%A A323689 _Paul D. Hanna_, Feb 14 2019

The rational summands are interpreted as formal power series over the complex numbers. Each positive-degree coefficient is a finite sum. The constant coefficient is replaced by 1 as the source requests; no value is assigned to an infinite sum of constants. The assertion includes an ordinary integer representative of each coefficient, with oddness equivalent to N = r^2 for a natural index.

Literature-screen completeness is ASSUMED-UNVERIFIED. Issue #21417 records the source revision and the unread dissertation and citation-screen gaps.

Exact-arithmetic fidelity evidence: the-omega-institute/trureturing-experiments, commit e0de2857d1d1616ee67af65221cc0028b888e4dd, docs/reports/oeis-a323689-hanna-square-parity/check.py, SHA256 b1a69c4f5dc4789d36755d31f212795fcb209457a7e947e60f8b1f8bf94305a4. The preregistration records exit 0 and ALL_CHECKS_OK; finite computations do not establish the unbounded assertion.