A056711 -id:A056711 - cp's OEIS frontend

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

R. J. Mathar, Jul 27 2016

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

			       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

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

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

T(2n,n) gives A292668.

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

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 *)

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

A046342 Number of 3-bead necklaces where each bead is a planted trivalent plane tree [or anything else enumerated by the Catalan numbers], by total number of nodes.

Original entry on oeis.org

1, 1, 3, 8, 24, 74, 245, 815, 2796, 9707, 34186, 121562, 436298, 1577310, 5740299, 21008777, 77279892, 285544700, 1059332082, 3944254118, 14734260864, 55207053787, 207421476390, 781283558998, 2949675307082, 11160264942376, 42309912978708, 160700303600030

Offset: 0

N. J. A. Sloane, Jan 19 2001

nonn

With offset = 3, a(n) is the number of forests having exactly three rooted plane trees with n total nodes. - Geoffrey Critzer, Feb 22 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000

See A058855 (a 6-bead analog) for details.

Cf. A000108, A058855, A056711, A275431.

nn=30;Drop[CoefficientList[Series[ CycleIndex[SymmetricGroup[3],s]/.Table[s[i]->(1-(1-4x^i)^(1/2))/2,{i,1,nn}],{x,0,nn}],x],3] (* Geoffrey Critzer, Feb 22 2013 *)

Plug g.f. for A000108, 1/2*(1-(1-4*x)^(1/2))/x, into cycle index for dihedral group D_6.
Cycle index for D_6: 1/6*Z[1]^3+1/2*Z[1]*Z[2]+1/3*Z[3].
a(n) = Sum_{j=0..3} A275431(n,j). - Alois P. Heinz, Sep 20 2017

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

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