A289978 Triangle read by rows: the multiset transform of the balanced binary Lyndon words (A022553).

1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 8, 4, 1, 1, 0, 25, 11, 4, 1, 1, 0, 75, 39, 12, 4, 1, 1, 0, 245, 124, 42, 12, 4, 1, 1, 0, 800, 431, 138, 43, 12, 4, 1, 1, 0, 2700, 1470, 490, 141, 43, 12, 4, 1, 1, 0, 9225, 5160, 1704, 504, 142, 43, 12, 4, 1, 1, 0, 32065, 18160, 6088, 1763, 507, 142, 43, 12, 4, 1, 1

Offset

0,8

Links

Alois P. Heinz, Rows n = 0..140, flattened

K. T. Chen, R. T. Fox and R. C. Lyndon, Free differential calculus IV. The quotient groups of the lower central series, Ann. Math. 68 (1) (1958) 81-95.

J.-P. Duval, Factorizing words over an ordered Alphabet, J. Algorithms 4 (4) (1983) 363.

R. J. Mathar, A bijection of Dyck Paths and multisets of Balanced Binary Lyndon Words (2021)

Index entries for sequences related to Lyndon words

Index entries for triangles generated by the Multiset Transformation

Formula

G.f.: Product_{j>=1} 1/(1-y*x^j)^A022553(j). - Alois P. Heinz, Jul 25 2017

Example

The triangle begins in row 0 and column 0 as:
1;
0       1;
0       1      1;
0       3      1      1;
0       8      4      1     1;
0      25     11      4     1     1;
0      75     39     12     4     1    1;
0     245    124     42    12     4    1    1;
0     800    431    138    43    12    4    1   1;
0    2700   1470    490   141    43   12    4   1   1;
0    9225   5160   1704   504   142   43   12   4   1  1;
0   32065  18160   6088  1763   507  142   43  12   4  1  1;
0  112632  64765  21790  6337  1777  508  142  43  12  4  1 1;
0  400023 232347  78845 22798  6396 1780  508 142  43 12  4 1 1;
0 1432613 840285 286652 82941 23047 6410 1781 508 142 43 12 4 1 1;

Maple

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(
       mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n))
    end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(binomial(g(i)+j-1, j)*
         b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Jul 25 2017

Mathematica

g[n_]:=g[n]=If[n==0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/(2n)]; b[n_, i_, p_]:=b[n, i, p]=If[p>n, 0, If[n==0, 1, If[Min[i, p]<1, 0, Sum[Binomial[g[i] + j - 1, j] b[n - i*j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]]; Table[b[n, n, k], {n, 0, 14}, {k, 0, n}]//Flatten (* Indranil Ghosh, Aug 05 2017, after Maple code *)
nn = 14;
b[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/ (2n)];
CoefficientList[#, y]& /@ (Series[Product[1/(1 - y x^i)^b[i], {i, 1, nn}], {x, 0, nn}] // CoefficientList[#, x]&) // Flatten (* Jean-François Alcover, Oct 29 2021 *)

Crossrefs

Cf. A022553 (column k=1), A000108 (row sums), A033184, A290277.

T(2n,n) gives A292287.

Sequence in context: A120060 A143295 A330892 * A185983 A179742 A285000

Adjacent sequences: A289975 A289976 A289977 * A289979 A289980 A289981

Keyword

nonn,tabl

Author

R. J. Mathar, Jul 18 2017

Status

approved