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

%I A275431 #23 May 18 2018 04:53:54
%S A275431 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,
%T A275431 1,1430,1008,358,105,28,8,2,1,4862,3536,1309,378,105,28,8,2,1,16796,
%U A275431 12618,4772,1410,384,105,28,8,2,1,58786,45220,17556,5220,1435,384,105,28,8,2,1
%N A275431 Triangle read by rows: T(n,k) = number of ways to insert n pairs of parentheses in k words.
%C A275431 Multiset transformation of A000108. Each word is dissected by a number of parentheses associated to its length.
%C A275431 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
%H A275431 Alois P. Heinz, <a href="/A275431/b275431.txt">Rows n = 1..141, flattened</a>
%H A275431 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A275431 T(n,1) = A000108(n).
%F A275431 T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1<k<=n.
%F A275431 G.f.: Product_{j>=1} 1/(1-y*x^j)^A000108(j). - _Alois P. Heinz_, Apr 13 2017
%e A275431        1
%e A275431        2     1
%e A275431        5     2     1
%e A275431       14     8     2     1
%e A275431       42    24     8     2     1
%e A275431      132    85    28     8     2     1
%e A275431      429   286   100    28     8     2     1
%e A275431     1430  1008   358   105    28     8     2     1
%e A275431     4862  3536  1309   378   105    28     8     2     1
%e A275431    16796 12618  4772  1410   384   105    28     8     2     1
%e A275431    58786 45220 17556  5220  1435   384   105    28     8     2     1
%p A275431 C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A275431 b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A275431       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
%p A275431        binomial(C(i)+j-1, j), j=0..min(n/i, p)))))
%p A275431     end:
%p A275431 T:= (n, k)-> b(n$2, k):
%p A275431 seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Apr 13 2017
%t A275431 c[n_] := c[n] = Binomial[2*n, n]/(n + 1);
%t A275431 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 A275431 T[n_, k_] := b[n, n, k];
%t A275431 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 18 2018, after _Alois P. Heinz_ *)
%Y A275431 Cf. A000108 (1st column), A007223 (2nd column), A056711 (3rd column), A088327 (row sums).
%Y A275431 T(2n,n) gives A292668.
%K A275431 nonn,tabl
%O A275431 1,2
%A A275431 _R. J. Mathar_, Jul 27 2016