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%I #8 Apr 08 2018 21:50:44
%S 1,1,1,1,7,1,29,18,1,95,288,1,275,2484,1080,1,742,15589,29748,1,1918,
%T 80269,420132,142800,1,4818,360801,4122572,5833728,1,11850,1467921,
%U 31844420,118722528,33747840,1,28655,5531163,207081545,1633525036,1869724800,1,68299,19603683,1181340677,17259989516,50714812224,12573792000
%N 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)).
%H 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.
%F Theorem 3.4 of Chen et al. (2018) gives a formula.
%e Triangle begins:
%e 1,
%e 1,1,
%e 1,7,
%e 1,29,18,
%e 1,95,288,
%e 1,275,2484,1080,
%e 1,742,15589,29748,
%e ...
%p s1 := proc(n,k)
%p (-1)^n*combinat[stirling1](n,k) ;
%p end proc:
%p A301731 := proc(n,k)
%p local a,m,i ;
%p a := 4*s1(n+1,k-1) ;
%p for m from 1 to n do
%p 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) ;
%p end do:
%p a/n/(n+1) ;
%p abs(%/2) ;
%p end proc:
%p for n from 1 to 15 do
%p for k from n+1 to 1 by -2 do
%p printf("%a,",A301731(n,k)) ;
%p end do:
%p printf("\n") ;
%p end do: # _R. J. Mathar_, Apr 08 2018
%K nonn,tabf
%O 1,5
%A _N. J. A. Sloane_, Apr 03 2018
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