A301731 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)).

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

Offset

1,5

Links

Table of n, a(n) for n=1..48.

Yichao Chen et al., On the genus distribution of wheels ..., Discrete Math., 341 (2018), 934-935.

Formula

Theorem 3.4 of Chen et al. (2018) gives a formula.

Example

Triangle begins:
1,
1,1,
1,7,
1,29,18,
1,95,288,
1,275,2484,1080,
1,742,15589,29748,
...

Maple

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

Crossrefs

Sequence in context: A146996 A286895 A083994 * A228498 A084181 A002678

Adjacent sequences: A301728 A301729 A301730 * A301732 A301733 A301734

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 03 2018

Status

approved