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

1,2

Comments

See Mullin (1967) for precise definition.

What is the sequence 1, 8, 72, 720, 7780, 89040, 1064644, 13173216, 167522976, 2178520080, ... in the leading diagonal?

Links

Table of n, a(n) for n=1..45.

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]

Formula

T(n,k) = (k+1)*A260040(n,k), n>=1, 0<=k<n.

Conjecture: T(n,0) = n*A168452(n-1). - R. J. Mathar, Jul 22 2015

Example

Triangle begins:
    1;
    8,   2;
   72,  30,  3;
  720, 380, 72, 4;
  ...

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 *)

Crossrefs

Row sums are A046715. Cf. A260040.

Sequence in context: A096041 A202625 A038280 * A032761 A262896 A188898

Adjacent sequences: A260036 A260037 A260038 * A260040 A260041 A260042

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jul 22 2015

Status

approved