bibkey: hanna2016diagonalindex authors: Paul D. Hanna year: 2016 title: “OEIS A266489, A300732, A300733, A292394, and A300734: vanishing-diagonal index divisibility” doi: null url: https://oeis.org/A266489 claim: “A266489: G.f. A(x) satisfies: [x^n] A( x/A(x)^n ) = 0 for n>1. (C1) n divides a(n) for n>=1: A268293(n) = a(n)/n. A300732: G.f. A(x) satisfies: [x^n] A( x/A(x)^(2n) ) = 0 for n>=1. Conjecture: n divides a(n) for n>=1. A300733: G.f. A(x) satisfies: [x^n] A( x/A(x)^(3n) ) = 0 for n>=1. Conjecture: n divides a(n) for n>=1. A292394: G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1. a(n) is divisible by n^2 for n>=1 (conjecture): A292395(n) = a(n)/n^2. A300734: G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n^2) ) = 0 for n>1. Conjecture: n^2 divides a(n) for n>=1.” strata_touched:
- D5/S1/Recurrence/Residue/DiagonalVanishingIndexDivisibility license: citation-only triage: anchor
Vanishing-diagonal index divisibility
Paul D. Hanna’s entries specify the five generating equations and divisibility
conjectures quoted above. A266489 is dated February 7, 2016; A292394 is dated
September 15, 2017; A300732, A300733, and A300734 are dated March 11, 2018.
The published data in all five entries begin with constant and linear
coefficients 1,1.
The normalized family uses a(0)=a(1)=1 and
[x^n] A(x/A(x)^e(n)) = 0 for n > 1. The literal NAMEs of A300732 and
A300733 instead print n>=1. At degree one, substitution by a series with
linear coefficient one preserves a(1)=1, so that printed boundary is
incompatible with their own data. The corresponding formal instances use
n>1; they do not assert the inconsistent degree-one vanishing equation.
The exponent functions are n, 2*n, 3*n, n^2, and 2*n^2. A single
integer-series construction proves the normalized generating equation and
uniqueness. For every natural k, if n^k divides e(n) for every natural
n, then n^k divides a(n) for every n>=1. Formal differentiation of
integer powers of a unit gives the exponent factor; strong induction and
a two-case comparison of prime multiplicities provide the divisibility of
each summand in the triangular coefficient equation.
At e(n)=n, uniqueness identifies the constructed series with
NegativePowerDiagonalModPrime.generatingSeries 2. Consequently A266489’s
clause C1 is proved for that existing coefficient function. Its separate
clause C2 is not part of the general index-divisibility assertion here.
Verified locator
- URL: https://oeis.org/A266489
- URL: https://oeis.org/A300732
- URL: https://oeis.org/A300733
- URL: https://oeis.org/A292394
- URL: https://oeis.org/A300734