A275431 Triangle read by rows: T(n,k) = number of ways to insert n pairs of parentheses in k words.

1, 2, 1, 5, 2, 1, 14, 8, 2, 1, 42, 24, 8, 2, 1, 132, 85, 28, 8, 2, 1, 429, 286, 100, 28, 8, 2, 1, 1430, 1008, 358, 105, 28, 8, 2, 1, 4862, 3536, 1309, 378, 105, 28, 8, 2, 1, 16796, 12618, 4772, 1410, 384, 105, 28, 8, 2, 1, 58786, 45220, 17556, 5220, 1435, 384, 105, 28, 8, 2, 1

Offset: 1

Author

R. J. Mathar, Jul 27 2016

Keywords

• nonn
• tabl

Comments

Multiset transformation of A000108. Each word is dissected by a number of parentheses associated to its length.

Also the number of forests of exactly k (unlabeled) ordered rooted trees with a total of n non-root nodes where each tree has at least 1 non-root node. - Alois P. Heinz, Sep 20 2017

Examples

			       1
       2     1
       5     2     1
      14     8     2     1
      42    24     8     2     1
     132    85    28     8     2     1
     429   286   100    28     8     2     1
    1430  1008   358   105    28     8     2     1
    4862  3536  1309   378   105    28     8     2     1
   16796 12618  4772  1410   384   105    28     8     2     1
   58786 45220 17556  5220  1435   384   105    28     8     2     1
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Index entries for triangles generated by the Multiset Transformation

Crossrefs

Cf. A000108 (1st column), A007223 (2nd column), A056711 (3rd column), A088327 (row sums).

T(2n,n) gives A292668.

Programs

Maple

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(C(i)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 13 2017

Mathematica

c[n_] := c[n] = Binomial[2*n, n]/(n + 1);
b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[c[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 18 2018, after Alois P. Heinz *)

Formula

T(n,1) = A000108(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1

G.f.: Product_{j>=1} 1/(1-y*x^j)^A000108(j). - Alois P. Heinz, Apr 13 2017

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