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A260039 Triangle read by rows giving numbers B(n,k) arising in the enumeration of doubly rooted tree maps.

%I A260039 #30 May 08 2023 09:34:33
%S A260039 1,8,2,72,30,3,720,380,72,4,7780,4690,1245,140,5,89040,58254,19152,
%T A260039 3192,240,6,1064644,734496,279972,60648,7000,378,7,13173216,9416688,
%U A260039 3997584,1046832,162000,13752,560,8,167522976,122687334,56488950,17086608,3285990,382140,24885,792,9
%N A260039 Triangle read by rows giving numbers B(n,k) arising in the enumeration of doubly rooted tree maps.
%C A260039 See Mullin (1967) for precise definition.
%C A260039 What is the sequence 1, 8, 72, 720, 7780, 89040, 1064644, 13173216, 167522976, 2178520080, ... in the leading diagonal?
%H A260039 R. C. Mullin, <a href="/A260039/a260039.pdf">On the average activity of a spanning tree of a rooted map</a>, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy] <a href="http://dx.doi.org/10.1016/S0021-9800(67)80001-2">[DOI]</a>
%F A260039 T(n,k) = (k+1)*A260040(n,k), n>=1, 0<=k<n.
%F A260039 Conjecture: T(n,0) = n*A168452(n-1). - _R. J. Mathar_, Jul 22 2015
%e A260039 Triangle begins:
%e A260039     1;
%e A260039     8,   2;
%e A260039    72,  30,  3;
%e A260039   720, 380, 72, 4;
%e A260039   ...
%p A260039 bEq64 := proc(k,u)
%p A260039     (k+1)*(2*u+k)!*(2*u+k+2)!/u!/(u+k+2)!/(u+k+1)!/(u+1)! ;
%p A260039 end proc:
%p A260039 Eq65 := proc(n,k)
%p A260039     add( bEq64(k,u)*bEq64(k,n-k-1-u),u=0..n-k-1) ;
%p A260039 end proc:
%p A260039 B := proc(n,k)
%p A260039     n*Eq65(n,k) ;
%p A260039 end proc:
%p A260039 for n from 1 to 10 do
%p A260039     for k from 0 to n-1 do
%p A260039         printf("%a,",B(n,k)) ;
%p A260039     end do:
%p A260039     printf("\n") ;
%p A260039 end do: # _R. J. Mathar_, Jul 22 2015
%t A260039 bEq64 [k_, u_] := (k + 1)*(2u + k)!*(2u + k + 2)!/u!/(u + k + 2)!/(u + k + 1)!/(u + 1)!;
%t A260039 Eq65[n_, k_] := Sum[bEq64[k, u]*bEq64[k, n - k - 1 - u], {u, 0, n - k - 1}];
%t A260039 B[n_, k_] := n*Eq65[n, k];
%t A260039 Table[B[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, May 08 2023, after _R. J. Mathar_ *)
%Y A260039 Row sums are A046715. Cf. A260040.
%K A260039 nonn,tabl
%O A260039 1,2
%A A260039 _N. J. A. Sloane_, Jul 22 2015