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Invert transform applied twice to Pascal's triangle A007318.
2

%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