A055373 Invert transform applied twice to Pascal's triangle A007318.

1, 1, 1, 3, 6, 3, 9, 27, 27, 9, 27, 108, 162, 108, 27, 81, 405, 810, 810, 405, 81, 243, 1458, 3645, 4860, 3645, 1458, 243, 729, 5103, 15309, 25515, 25515, 15309, 5103, 729, 2187, 17496, 61236, 122472, 153090, 122472, 61236, 17496, 2187, 6561

Offset: 0

Author

Christian G. Bower, May 16 2000

Keywords

• nonn
• tabl

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 2, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of sequences of nonempty sequences of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence of sequences. In other words, these are sequences of the structures counted by A055372. - Geoffrey Critzer, Apr 06 2013

Examples

			Triangle begins:
   1;
   1,   1;
   3,   6,   3;
   9,  27,  27,   9;
  27, 108, 162, 108,  27;
  ...
		

Links

• N. J. A. Sloane, Transforms
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A000244, A007318, A055372, A055374.

Cf. A084938.

Programs

Mathematica

nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));b=1/(1-a);Map[f,CoefficientList[Series[1/(2-b),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Apr 06 2013 *)

Formula

T(n,k) = 3^(n-1)*C(n, k) for n > 0.

O.g.f.: 1/(2 - A(x,y)) where A(x,y) is the o.g.f. for A055372. - Geoffrey Critzer, Apr 06 2013