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Invert transform of Pascal's triangle A007318.
14

%I #32 Jan 21 2020 09:04:13

%S 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 640,480,192,32,64,448,1344,2240,2240,1344,448,64,128,1024,3584,7168,

%U 8960,7168,3584,1024,128,256,2304,9216,21504,32256,32256,21504,9216,2304,256

%N Invert transform of Pascal's triangle A007318.

%C 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 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 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 a(n,k) = 2^(n-1)*C(n, k), for n>0.

%F G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).

%F As an infinite lower triangular matrix, equals A134309 * A007318. - _Gary W. Adamson_, Oct 19 2007

%F 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 Triangle begins:

%e 1;

%e 1, 1;

%e 2, 4, 2;

%e 4, 12, 12, 4;

%e 8, 32, 48, 32, 8;

%e ...

%t 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 Row sums give A081294. Cf. A000079, A007318, A055373, A055374.

%Y Cf. A134309.

%Y T(2n,n) gives A098402.

%K nonn,tabl

%O 0,4

%A _Christian G. Bower_, May 16 2000