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A055372
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Invert transform of Pascal's triangle A007318.
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11
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 4, 2;
4, 12, 12, 4;
8, 32, 48, 32, 8;
...
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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Christian G. Bower, May 16 2000
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STATUS
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approved
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