Revision History for A301731
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| A301731
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Triangle read by rows: T(n,m) = one-half of number of embeddings of the n-wheel with k = n+1-2m regions (n >= 1, 0 <= m <= floor(n/2)).
(history;
published version)
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#8 by R. J. Mathar at Sun Apr 08 21:50:44 EDT 2018
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#7 by R. J. Mathar at Sun Apr 08 21:49:23 EDT 2018
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| DATA
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1, 1, 1, 1, 7, 1, 29, 18, 1, 95, 288, 1, 275, 2484, 1080, 1, 742, 15589, 29748, 1, 1918, 80269, 420132, 142800, 1, 4818, 360801, 4122572, 5833728, 1, 11850, 1467921, 31844420, 118722528, 33747840, 1, 28655, 5531163, 207081545, 1633525036, 1869724800, 1, 68299, 19603683, 1181340677, 17259989516, 50714812224, 12573792000, 1, 160745, 66077583, 6076248035, 150181724836, 914389931520, 891276410880, 1, 374101, 213560347, 28728259703, 1124236561448, 12461968618736, 30595454901504, 6801152117760
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#6 by R. J. Mathar at Sun Apr 08 21:49:01 EDT 2018
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| DATA
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1, 1, 1, 1, 7, 1, 29, 18, 1, 95, 288, 1, 275, 2484, 1080, 1, 742, 15589, 29748, 1, 1918, 80269, 420132, 142800, 1, 4818, 360801, 4122572, 5833728, 1, 11850, 1467921, 31844420, 118722528, 33747840, 1, 28655, 5531163, 207081545, 1633525036, 1869724800, 1, 68299, 19603683, 1181340677, 17259989516, 50714812224, 12573792000, 1, 160745, 66077583, 6076248035, 150181724836, 914389931520, 891276410880, 1, 374101, 213560347, 28728259703, 1124236561448, 12461968618736, 30595454901504, 6801152117760
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| MAPLE
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s1 := proc(n, k)
(-1)^n*combinat[stirling1](n, k) ;
end proc:
A301731 := proc(n, k)
local a, m, i ;
a := 4*s1(n+1, k-1) ;
for m from 1 to n do
a := a+add( binomial(k+i, k) *binomial(n, m) *((n+1)^i-(n+1-m)^i-(m+1)^i+1) *s1(n+2, k+1+i), i=0..n+1-k) ;
end do:
a/n/(n+1) ;
abs(%/2) ;
end proc:
for n from 1 to 15 do
for k from n+1 to 1 by -2 do
printf("%a, ", A301731(n, k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Apr 08 2018
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| KEYWORD
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nonn,tabf,more,new
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| STATUS
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approved
editing
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#5 by R. J. Mathar at Sun Apr 08 01:17:45 EDT 2018
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#4 by R. J. Mathar at Sun Apr 08 01:17:34 EDT 2018
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| REFERENCES
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Yichao Chen et al., On the genus distribution of wheels ..., Discrete Math., 341 (2018), 934-935.
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| LINKS
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Yichao Chen et al., <a href="http://doi.org/10.1016/j.disc.2017.12.007">On the genus distribution of wheels ...</a>, Discrete Math., 341 (2018), 934-935.
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approved
editing
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#3 by N. J. A. Sloane at Tue Apr 03 11:51:50 EDT 2018
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#2 by N. J. A. Sloane at Tue Apr 03 11:42:27 EDT 2018
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| NAME
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allocated for N. J. A. Sloane
Triangle read by rows: T(n,m) = one-half of number of embeddings of the n-wheel with k = n+1-2m regions (n >= 1, 0 <= m <= floor(n/2)).
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| DATA
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1, 1, 1, 1, 7, 1, 29, 18, 1, 95, 288, 1, 275, 2484, 1080, 1, 742, 15589, 29748
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| OFFSET
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1,5
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| REFERENCES
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Yichao Chen et al., On the genus distribution of wheels ..., Discrete Math., 341 (2018), 934-935.
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| FORMULA
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Theorem 3.4 of Chen et al. (2018) gives a formula.
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| EXAMPLE
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Triangle begins:
1,
1,1,
1,7,
1,29,18,
1,95,288,
1,275,2484,1080,
1,742,15589,29748,
...
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| KEYWORD
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allocated
nonn,tabf,more
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| AUTHOR
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N. J. A. Sloane, Apr 03 2018
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| STATUS
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approved
editing
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#1 by N. J. A. Sloane at Sun Mar 25 22:07:13 EDT 2018
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| NAME
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allocated for N. J. A. Sloane
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| KEYWORD
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allocated
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| STATUS
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approved
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