A253180 - OEIS

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.

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

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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:

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