Metadata
bibkey: hanna2013derivativesubstitutioncongruences
authors: Paul D. Hanna
year: 2013
title: Derivative-substitution exponential generating functions A232694, A232695, A232696 and A232686
doi: null
url: https://oeis.org/A232694
claim: The nominated COMMENTS conjectures concern congruences of the factorial-normalized coefficients of four derivative-substitution exponential generating functions.
strata_touched:
- D5/S1/Recurrence/Residue/HannaEGFMachinery
- D5/S1/Recurrence/Residue/HannaA232694Congruences
- D5/S1/Recurrence/Residue/HannaA232695Congruences
- D5/S1/Recurrence/Residue/HannaA232696Congruences
- D5/S1/Recurrence/Residue/HannaA232686Congruences
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A232694
Related source records: https://oeis.org/A232695, https://oeis.org/A232696, https://oeis.org/A232686. OEIS records have no page numbering and no DOI or arXiv identifier.
Source statements
A232694
%I A232694 #7 Dec 03 2013 20:49:42
%N A232694 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x)^4) with A(0)=1.
%C A232694 CONJECTURES.
%C A232694 a(n) == 0 (mod 3) for n>=3.
%C A232694 a(n) == 1 (mod 8) for n>=0.
%e A232694 E.g.f.: A(x) = 1 + x + x^2/2! + 9*x^3/3! + 177*x^4/4! + 5601*x^5/5! +...
%O A232694 0,4
%A A232694 _Paul D. Hanna_, Nov 27 2013
A232695
%I A232695 #22 Dec 03 2013 20:44:17
%N A232695 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x)^5) with A(0)=1.
%C A232695 CONJECTURES.
%C A232695 a(n) == 1 (mod 5) for n>=0.
%C A232695 a(n) == 0 (mod 2) for n>=4.
%C A232695 a(n) == 0 (mod 2^2) for n>=10.
%C A232695 a(n) == 0 (mod 2^3) for n>=18.
%C A232695 a(n) == 0 (mod 2^k) for n>=(8*k-6) for k>1.
%e A232695 E.g.f.: A(x) = 1 + x + x^2/2! + 11*x^3/3! + 266*x^4/4! + 10326*x^5/5! +...
%O A232695 0,4
%A A232695 _Paul D. Hanna_, Nov 27 2013
A232696
%I A232696 #11 Dec 03 2013 20:51:28
%N A232696 E.g.f. A(x) satisfies: A'(x) = A(x/A'(x)) with A(0)=1.
%C A232696 CONJECTURES.
%C A232696 a(n) == 0 (mod 2) for n>=4.
%C A232696 a(n) == 0 (mod 2^2) for n>=10.
%C A232696 a(n) == 0 (mod 2^3) for n>=18.
%C A232696 a(n) == 0 (mod 2^k) for n>=(8*n-6) for k>1.
%e A232696 E.g.f.: A(x) = 1 + x + x^2/2! - x^3/3! + 2*x^4/4! + 6*x^5/5! - 264*x^6/6! + 5370*x^7/7! +...
%O A232696 0,5
%A A232696 _Paul D. Hanna_, Nov 28 2013
A232686
%I A232686 #16 Dec 07 2013 12:29:18
%N A232686 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x))^2.
%C A232686 CONJECTURES.
%C A232686 a(n) == 0 (mod 2) for n>=2.
%C A232686 a(n) == 0 (mod 2^2) for n>=4.
%C A232686 a(n) == 0 (mod 2^3) for n>=5.
%C A232686 a(n) == 0 (mod 2^k) for n>=(2*k-1), k>=3.
%C A232686 a(2*n) == 2^n (mod 2^(n+1)) for n>=0.
%e A232686 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 196*x^4/4! + 4352*x^5/5! +...
%O A232686 0,3
%A A232686 _Paul D. Hanna_, Dec 03 2013
Interpretation and scope
The coefficient a(n) is n! times the ordinary rational coefficient of x^n. The initial coefficient is one in every displayed exponential series. The selected pairs are modulo 3 and 8 for A232694, modulo 5 and 2 for A232695, modulo 2 and 4 for A232696, and modulo 2 and 4 for A232686. The other quoted conjectures are outside these four propositions. The reciprocal equation is interpreted by formal unit inversion, with derivative constant coefficient one.
Evidence and literature boundary
The exact-coefficient experiment is in the private repository
the-omega-institute/trureturing-experiments, commit
e2566e824df32e3f308cd1b801ef82dd17fa64cf,
docs/reports/hanna-efg-congruences/check.py, SHA-256
14fa66db49524fd5b8ccbf45b2245d0a4813980bff076b1b0a855d869393f259.
The recorded command is python3 check.py > output.txt, exit 0. It checks
indices 0 through 300 and compares independent rational-series iteration through degree 12.
These finite calculations support the coefficient interpretation; they do not prove an unbounded congruence.
The all-index conclusions are repository derivations from integral recurrences and exact residuals.
Literature-screen completeness is ASSUMED-UNVERIFIED; no prior settlement was located
in the sources inspected for issues #21485, #21486, #21487 and #21488.