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%I A055372 #32 Jan 21 2020 09:04:13
%S A055372 1,1,1,2,4,2,4,12,12,4,8,32,48,32,8,16,80,160,160,80,16,32,192,480,
%T A055372 640,480,192,32,64,448,1344,2240,2240,1344,448,64,128,1024,3584,7168,
%U A055372 8960,7168,3584,1024,128,256,2304,9216,21504,32256,32256,21504,9216,2304,256
%N A055372 Invert transform of Pascal's triangle A007318.
%C A055372 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Aug 10 2005
%C A055372 T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence.  For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - _Geoffrey Critzer_, Apr 06 2013
%H A055372 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A055372 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A055372 a(n,k) = 2^(n-1)*C(n, k), for n>0.
%F A055372 G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).
%F A055372 As an infinite lower triangular matrix, equals A134309 * A007318. - _Gary W. Adamson_, Oct 19 2007
%F A055372 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - _Philippe Deléham_, Feb 05 2012
%e A055372 Triangle begins:
%e A055372   1;
%e A055372   1,  1;
%e A055372   2,  4,  2;
%e A055372   4, 12, 12,  4;
%e A055372   8, 32, 48, 32,  8;
%e A055372   ...
%t A055372 nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));Map[f,CoefficientList[Series[1/(1-a),{x,0,nn}],{x,y}]]//Grid  (* _Geoffrey Critzer_, Apr 06 2013 *)
%Y A055372 Row sums give A081294. Cf. A000079, A007318, A055373, A055374.
%Y A055372 Cf. A134309.
%Y A055372 T(2n,n) gives A098402.
%K A055372 nonn,tabl
%O A055372 0,4
%A A055372 _Christian G. Bower_, May 16 2000

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