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A253180 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A253180 #28 Sep 28 2023 05:28:34
%S A253180 1,0,1,0,2,2,0,5,15,5,0,14,98,84,14,0,42,630,1050,420,42,0,132,4092,
%T A253180 11880,8580,1980,132,0,429,27027,129129,150150,60060,9009,429,0,1430,
%U A253180 181610,1381380,2432430,1501500,380380,40040,1430,0,4862,1239810,14707550,37777740,33795762,12864852,2246244,175032,4862
%N A253180 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A253180 In general, column k>0 is asymptotic to (4*k)^n / (k!*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%H A253180 Alois P. Heinz, <a href="/A253180/b253180.txt">Rows n = 0..140, flattened</a>
%F A253180 T(n,k) = A256061(n,k)/k! = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n) / A000142(n).
%e A253180 T(3,1) = 5: ()()(), ()(()), (())(), (()()), ((())).
%e A253180 T(3,2) = 15: ()()[], ()[](), ()[][], ()([]), ()[()], ()[[]], (())[], ([])(), ([])[], (()[]), ([]()), ([][]), (([])), ([()]), ([[]]).
%e A253180 T(3,3) = 5: ()[]{}, ()[{}], ([]){}, ([]{}), ([{}]).
%e A253180 Triangle T(n,k) begins:
%e A253180   1;
%e A253180   0,   1;
%e A253180   0,   2,     2;
%e A253180   0,   5,    15,      5;
%e A253180   0,  14,    98,     84,     14;
%e A253180   0,  42,   630,   1050,    420,    42;
%e A253180   0, 132,  4092,  11880,   8580,  1980,  132;
%e A253180   0, 429, 27027, 129129, 150150, 60060, 9009, 429;
%e A253180   ...
%p A253180 ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A253180 A:= proc(n, k) option remember; k^n*ctln(n) end:
%p A253180 T:= (n, k)-> add(A(n, k-i)*(-1)^i/((k-i)!*i!), i=0..k):
%p A253180 seq(seq(T(n, k), k=0..n), n=0..10);
%t A253180 A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[0, 0] = 1; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/((k-i)!*i!), {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 11 2017, adapted from Maple *)
%Y A253180 Columns k=0-10 give: A000007, A000108 (for n>0), A258390, A258391, A258392, A258393, A258394, A258395, A258396, A258397, A258398.
%Y A253180 Main diagonal gives A000108.
%Y A253180 First lower diagonal gives A002740(n+2).
%Y A253180 T(2n,n) gives A258399.
%Y A253180 Row sums give A064299.
%Y A253180 Cf. A000142, A256061.
%K A253180 nonn,tabl
%O A253180 0,5
%A A253180 _Alois P. Heinz_, Mar 23 2015