|
%I #30 Sep 28 2023 05:27:17
%S 1,0,1,0,2,4,0,5,30,30,0,14,196,504,336,0,42,1260,6300,10080,5040,0,
%T 132,8184,71280,205920,237600,95040,0,429,54054,774774,3603600,
%U 7207200,6486480,2162160,0,1430,363220,8288280,58378320,180180000,273873600,201801600,57657600
%N 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.
%C Also number of binary trees with n inner nodes of exactly k different dimensions. T(2,2) = 4:
%C : balanced parentheses : ([]) : [()] : ()[] : []() :
%C :----------------------:-------:-------:-------:-------:
%C : trees : (1) : [2] : (1) : [2] :
%C : : / \ : / \ : / \ : / \ :
%C : : [2] : (1) : [2] : (1) :
%C : : / \ : / \ : / \ : / \ :
%H Alois P. Heinz, <a href="/A256061/b256061.txt">Rows n = 0..140, flattened</a>
%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n).
%F T(n,k) = k! * A253180(n,k).
%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A290605(n,k-i). - _Alois P. Heinz_, Oct 28 2019
%e A(3,2) = 30: (())[], (()[]), (([])), ()()[], ()([]), ()[()], ()[[]], ()[](), ()[][], ([()]), ([[]]), ([]()), ([])(), ([])[], ([][]), [(())], [()()], [()[]], [()](), [()][], [([])], [[()]], [[]()], [[]](), [](()), []()(), []()[], []([]), [][()], [][]().
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2, 4;
%e 0, 5, 30, 30;
%e 0, 14, 196, 504, 336;
%e 0, 42, 1260, 6300, 10080, 5040;
%e 0, 132, 8184, 71280, 205920, 237600, 95040;
%e 0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160;
%e ...
%p ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A:= proc(n, k) option remember; k^n*ctln(n) end:
%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p seq(seq(T(n, k), k=0..n), n=0..10);
%t 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 *)
%Y Columns k=0-1 give: A000007, A000108 (for n>0).
%Y Main diagonal gives A001761.
%Y Cf. A253180, A255982, A258427, A290605.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Mar 13 2015
|
