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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. 15
 1, 0, 1, 0, 2, 2, 0, 5, 15, 5, 0, 14, 98, 84, 14, 0, 42, 630, 1050, 420, 42, 0, 132, 4092, 11880, 8580, 1980, 132, 0, 429, 27027, 129129, 150150, 60060, 9009, 429, 0, 1430, 181610, 1381380, 2432430, 1501500, 380380, 40040, 1430, 0, 4862, 1239810, 14707550, 37777740, 33795762, 12864852, 2246244, 175032, 4862 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS In general, column k>0 is asymptotic to (4*k)^n / (k!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,k) = A256061(n,k)/k! = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n) / A000142(n). EXAMPLE T(3,1) = 5: ()()(), ()(()), (())(), (()()), ((())). T(3,2) = 15: ()()[], ()[](), ()[][], ()([]), ()[()], ()[[]], (())[], ([])(), ([])[], (()[]), ([]()), ([][]), (([])), ([()]), ([[]]). T(3,3) = 5: ()[]{}, ()[{}], ([]){}, ([]{}), ([{}]). Triangle T(n,k) begins:   1;   0,   1;   0,   2,     2;   0,   5,    15,      5;   0,  14,    98,     84,     14;   0,  42,   630,   1050,    420,    42;   0, 132,  4092,  11880,   8580,  1980,  132;   0, 429, 27027, 129129, 150150, 60060, 9009, 429; 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/((k-i)!*i!), i=0..k): seq(seq(T(n, k), k=0..n), n=0..10); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 give: A000007, A000108 (for n>0), A258390, A258391, A258392, A258393, A258394, A258395, A258396, A258397, A258398. Main diagonal gives A000108. First lower diagonal gives A002740(n+2). T(2n,n) gives A258399. Row sums give A064299. Cf. A000142, A256061. Sequence in context: A277295 A254749 A129936 * A295214 A221408 A221396 Adjacent sequences:  A253177 A253178 A253179 * A253181 A253182 A253183 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 23 2015 STATUS approved

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Last modified May 17 22:15 EDT 2021. Contains 343992 sequences. (Running on oeis4.)