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A256061 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 6
1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 14, 196, 504, 336, 0, 42, 1260, 6300, 10080, 5040, 0, 132, 8184, 71280, 205920, 237600, 95040, 0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160, 0, 1430, 363220, 8288280, 58378320, 180180000, 273873600, 201801600, 57657600 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Also number of binary trees with n inner nodes of exactly k different dimensions. T(2,2) = 4:
: balanced parentheses : ([]) : [()] : ()[] : []() :
:----------------------:-------:-------:-------:-------:
: trees : (1) : [2] : (1) : [2] :
: : / \ : / \ : / \ : / \ :
: : [2] : (1) : [2] : (1) :
: : / \ : / \ : / \ : / \ :
LINKS
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n).
T(n,k) = k! * A253180(n,k).
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A290605(n,k-i). - Alois P. Heinz, Oct 28 2019
EXAMPLE
A(3,2) = 30: (())[], (()[]), (([])), ()()[], ()([]), ()[()], ()[[]], ()[](), ()[][], ([()]), ([[]]), ([]()), ([])(), ([])[], ([][]), [(())], [()()], [()[]], [()](), [()][], [([])], [[()]], [[]()], [[]](), [](()), []()(), []()[], []([]), [][()], [][]().
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 4;
0, 5, 30, 30;
0, 14, 196, 504, 336;
0, 42, 1260, 6300, 10080, 5040;
0, 132, 8184, 71280, 205920, 237600, 95040;
0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160;
...
MAPLE
ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
A[0, 0] = 1; A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)
CROSSREFS
Columns k=0-1 give: A000007, A000108 (for n>0).
Main diagonal gives A001761.
Sequence in context: A269011 A274086 A255982 * A323099 A002652 A202541
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 13 2015
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)