A260039 Triangle read by rows giving numbers B(n,k) arising in the enumeration of doubly rooted tree maps.
1, 8, 2, 72, 30, 3, 720, 380, 72, 4, 7780, 4690, 1245, 140, 5, 89040, 58254, 19152, 3192, 240, 6, 1064644, 734496, 279972, 60648, 7000, 378, 7, 13173216, 9416688, 3997584, 1046832, 162000, 13752, 560, 8, 167522976, 122687334, 56488950, 17086608, 3285990, 382140, 24885, 792, 9
Offset: 1
Examples
Triangle begins:
1;
8, 2;
72, 30, 3;
720, 380, 72, 4;
...
Links
- R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy] [DOI]
Programs
-
Maple
bEq64 := proc(k,u) (k+1)*(2*u+k)!*(2*u+k+2)!/u!/(u+k+2)!/(u+k+1)!/(u+1)! ; end proc: Eq65 := proc(n,k) add( bEq64(k,u)*bEq64(k,n-k-1-u),u=0..n-k-1) ; end proc: B := proc(n,k) n*Eq65(n,k) ; end proc: for n from 1 to 10 do for k from 0 to n-1 do printf("%a,",B(n,k)) ; end do: printf("\n") ; end do: # R. J. Mathar, Jul 22 2015 -
Mathematica
bEq64 [k_, u_] := (k + 1)*(2u + k)!*(2u + k + 2)!/u!/(u + k + 2)!/(u + k + 1)!/(u + 1)!; Eq65[n_, k_] := Sum[bEq64[k, u]*bEq64[k, n - k - 1 - u], {u, 0, n - k - 1}]; B[n_, k_] := n*Eq65[n, k]; Table[B[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 08 2023, after R. J. Mathar *)
Formula
T(n,k) = (k+1)*A260040(n,k), n>=1, 0<=k
Conjecture: T(n,0) = n*A168452(n-1). - R. J. Mathar, Jul 22 2015
Comments