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

0,4

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

Links

Table of n, a(n) for n=0..54.

N. J. A. Sloane, Transforms

Index entries for triangles and arrays related to Pascal's triangle

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

Example

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

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 *)

Crossrefs

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

Cf. A134309.

T(2n,n) gives A098402.

Sequence in context: A240893 A241108 A151706 * A241078 A198285 A136620

Adjacent sequences: A055369 A055370 A055371 * A055373 A055374 A055375

Keyword

nonn,tabl

Author

Christian G. Bower, May 16 2000

Status

approved