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Revision History for A355024 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

Number of unlabeled trees on n nodes with maximum degree three and three vertices of degree three.
(history; published version)

#24 by Joerg Arndt at Mon Aug 22 04:49:31 EDT 2022

STATUS

reviewed

approved

#23 by Michel Marcus at Mon Aug 22 04:47:26 EDT 2022

STATUS

proposed

reviewed

#22 by Jon E. Schoenfield at Mon Aug 22 03:55:31 EDT 2022

STATUS

editing

proposed

#21 by Jon E. Schoenfield at Mon Aug 22 03:55:29 EDT 2022

LINKS

Marko R. Riedel et al., math.stackexchange.com, Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4472439/">Trees with maximum degree three and three vertices of degree three</a>.

STATUS

approved

editing

#20 by Peter Luschny at Thu Jun 16 15:57:06 EDT 2022

STATUS

proposed

approved

#19 by Marko Riedel at Thu Jun 16 13:09:56 EDT 2022

STATUS

editing

proposed

Discussion

Thu Jun 16

13:15

Marko Riedel: For n=29 the value is 46627 but 29^6/5670 ~ 103267.

13:34

Stefano Spezia: My formula is asymptotic formula: e.g. for n = 5000, the ratio a(n)/b(n) = 0.995807, with b(n) = n^6/5760.

13:35

Stefano Spezia: It is easy to prove that a(n)/b(n) -> 1 for n approaching infinity

13:53

Marko Riedel: Convert to partial fractions and extract binomial coefficients?

13:57

Marko Riedel: You get [z^n] 1/8 (1-z)^7 = 1/8 C(n, 6) ~ n^6/5760. I understand.

14:40

Stefano Spezia: Exactly

#18 by Marko Riedel at Thu Jun 16 13:03:54 EDT 2022

CROSSREFS

Cf. A355023.

STATUS

proposed

editing

#17 by Peter Luschny at Thu Jun 16 07:06:09 EDT 2022

STATUS

editing

proposed

#16 by Peter Luschny at Thu Jun 16 07:05:29 EDT 2022

MAPLE

gf := z^8*(1 - z + 2*z^2)/((1 - z)^7*(1 + z)^3*(1 + z^2)): ser := series(gf, z, 42): seq(coeff(ser, z, n), n = 8..40); # _Peter Luschny_, Jun 16 2022

ser := series(gf, z, 42): seq(coeff(ser, z, n), n = 8..40); # Peter Luschny, Jun 16 2022

#15 by Peter Luschny at Thu Jun 16 07:03:44 EDT 2022

FORMULA

G.f.: (1/8)*z^8*(1/(1-z)^7 + 2/((1-z)^5*(1-z^2)) + 1/((1-z)^3*(1-z^2)^2) + 2/((1-z)*(1-z^2)^3) + 2/((1-z)*(1-z^2)*(1-z^4))).

G.f.: z^8*(1 - z + 2*z^2)/((1 - z)^7*(1 + z)^3*(1 + z^2)).

MAPLE

gf := z^8*(1 - z + 2*z^2)/((1 - z)^7*(1 + z)^3*(1 + z^2)):

ser := series(gf, z, 42): seq(coeff(ser, z, n), n = 8..40); # Peter Luschny, Jun 16 2022

STATUS

proposed

editing