A133611 A triangular array of numbers related to factorization and number of parts in Murasaki diagrams.

1, 1, 1, 2, 2, 1, 5, 5, 4, 1, 15, 15, 14, 7, 1, 52, 52, 51, 36, 11, 1, 203, 203, 202, 171, 81, 16, 1, 877, 877, 876, 813, 512, 162, 22, 1, 4140, 4140, 4139, 4012, 3046, 1345, 295, 29, 1, 21147, 21147, 21146, 20891, 17866, 10096, 3145, 499, 37, 1, 115975, 115975, 115974, 115463, 106133, 72028, 29503, 6676, 796, 46, 1

Offset

1,4

Comments

When the Bell multisets are encoded as described in A130274, the seven case in the example can be coded as 19578, 15942, 30873, 26427, 35642, 29491 and 32938.

Links

Robert Israel, Table of n, a(n) for n = 1..10011

Formula

Equals A048993 * A000012. - Gary W. Adamson, Jan 29 2008

That is, T(i,j) = Sum_{k=j..i} A048993(i,k) for 0 <= j <= i. - Robert Israel, Nov 01 2018

Example

The array begins:
   1
   1  1
   2  2  1
   5  5  4  1
  15 15 14  7  1
  52 52 51 36 11 1
  ...
a(14) = 7 because only seven of the 52 Bell multisets can be generated by attaching a new stroke to the third element in the set of diaqrams with four strokes.

Maple

T:= proc(i, j) add(combinat:-stirling2(i, k), k=j..i) end proc:
seq(seq(T(i, j), j=0..i), i=0..15); # Robert Israel, Nov 01 2018
# second Maple program:
b:= proc(n, t) option remember; `if`(n>0, add(b(n-j, t+1)*
      binomial(n-1, j-1), j=1..n), add(x^j, j=0..t))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 30 2019

Mathematica

row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse;
Array[row, 11, 0] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Crossrefs

Cf. A000110 (row sums), A137650 (a similar triangle), A130274, A211561.

Cf. A048993.

Sequence in context: A185414 A346520 A362925 * A010094 A019710 A118806

Adjacent sequences: A133608 A133609 A133610 * A133612 A133613 A133614

Keyword

nonn,tabl

Author

Alford Arnold, Sep 18 2007

Extensions

Definition not clear to me - N. J. A. Sloane, Sep 18 2007

More terms from Robert Israel, Nov 01 2018

Status

approved