Generating functions that don't match

[Charles Greathouse]

I recently noticed that both A167978 and A167980 have two versions of their (rational ordinary) generating functions which don't match. Can anyone tell which are right?

I'm not terribly pleased to see this inconsistency in the OEIS.

[Robin Houston]

For A167978, the series expansions of the two given generating functions first differ at the x^17 term.

The first-listed g.f. gives a value that matches the one in the table at https://oeis.org/A167978/b167978.txt,

whereas the second-listed g.f. gives 18889617541497286590540380967.

Robin

On Mon, 18 May 2026 at 13:19, Charles Greathouse <crgrea...@gmail.com> wrote:

I recently noticed that both A167978 and A167980 have two versions of their (rational ordinary) generating functions which don't match. Can anyone tell which are right?

I'm not terribly pleased to see this inconsistency in the OEIS.

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[D. S. McNeil]

Maybe the author meant 1080 instead of 1081?

sage: long_num
x^16 + 2*x^15 + 2*x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1
sage: long_den
1035*x^16 - 45*x^15 - 45*x^14 - 45*x^13 - 45*x^12 - 45*x^11 - 45*x^10 - 45*x^9 - 45*x^8 - 45*x^7 - 45*x^6 - 45*x^5 - 45*x^4 - 45*x^3 - 45*x^2 - 45*x + 1
sage: (1+x)*(1-x^16) / (-1035*x^17 + 1080*x^16 - 46*x + 1) - long_num/long_den
0

The ^17 in the program also looks wrong to me.

Doug

[Christian Sievers]

Indeed, replace the 1081 with 1080 for A167978.

And for A167980, replace x^17 with x^16 in the numerator of the second g.f.

Christian

[Christian Sievers]

Oh wait, note that I'm just observing that that is a way to change the second g.f. so that it becomes equal to the first.

I don't know if that is the right one.

Christian

[D. S. McNeil]

To clear the ambiguity, FWIW I think I can confirm the original gf from the Steinberg formula:

sage: r = 47 # num gen
sage: m = 16 # coxeter exp
sage: C = binomial(r, 2)
sage: C
1081
sage: R.<x> = PolynomialRing(QQ)
sage: bracket = sum(x^i for i in range(m))
sage: long_num = (1 + x) * bracket
sage: long_den = 1  - (r - 2) * (bracket-1)  + binomial(r - 1, 2) * x^m
sage: long_num, long_den
(x^16 + 2*x^15 + 2*x^14 + 2*x^13 + 2*x^12 + 2*x^11 + 2*x^10 + 2*x^9 + 2*x^8 + 2*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1,
 1035*x^16 - 45*x^15 - 45*x^14 - 45*x^13 - 45*x^12 - 45*x^11 - 45*x^10 - 45*x^9 - 45*x^8 - 45*x^7 - 45*x^6 - 45*x^5 - 45*x^4 - 45*x^3 - 45*x^2 - 45*x + 1)
sage: compact_num, compact_den
(-x^17 - x^16 + x + 1, -1035*x^17 + 1080*x^16 - 46*x + 1)
sage: long_num/long_den - compact_num/compact_den
0

Doug

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[Charles Greathouse]

Thank you Doug!

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