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A260039 Triangle read by rows giving numbers B(n,k) arising in the enumeration of doubly rooted tree maps. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jul 22 2015
STATUS
approved

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Last modified March 4 05:23 EST 2024. Contains 370522 sequences. (Running on oeis4.)