A055372 Invert transform of Pascal's triangle A007318.

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

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, 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

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

Examples

			Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  4, 12, 12,  4;
  8, 32, 48, 32,  8;
  ...
		

Links

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

Crossrefs

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

Cf. A134309.

T(2n,n) gives A098402.

Programs

Mathematica

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 *)

Formula

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

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

As an infinite lower triangular matrix, equals A134309 * A007318. - Gary W. Adamson, Oct 19 2007

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