%I #34 Jan 19 2023 05:21:39
%S 1,1,1,3,6,3,9,27,27,9,27,108,162,108,27,81,405,810,810,405,81,243,
%T 1458,3645,4860,3645,1458,243,729,5103,15309,25515,25515,15309,5103,
%U 729,2187,17496,61236,122472,153090,122472,61236,17496,2187,6561
%N Invert transform applied twice to Pascal's triangle A007318.
%C 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
%C 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
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F T(n,k) = 3^(n-1)*C(n, k) for n > 0.
%F O.g.f.: 1/(2 - A(x,y)) where A(x,y) is the o.g.f. for A055372. - _Geoffrey Critzer_, Apr 06 2013
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 6, 3;
%e 9, 27, 27, 9;
%e 27, 108, 162, 108, 27;
%e ...
%t 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 *)
%Y Cf. A000244, A007318, A055372, A055374.
%Y Cf. A084938.
%K nonn,tabl
%O 0,4
%A _Christian G. Bower_, May 16 2000