login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A289978 Triangle read by rows: the multiset transform of the balanced binary Lyndon words (A022553). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

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 *)

CROSSREFS

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

T(2n,n) gives A292287.

Sequence in context: A115378 A120060 A143295 * A185983 A179742 A285000

Adjacent sequences:  A289975 A289976 A289977 * A289979 A289980 A289981

KEYWORD

nonn,tabl

AUTHOR

R. J. Mathar, Jul 18 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 21 03:06 EST 2019. Contains 320364 sequences. (Running on oeis4.)