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A008949
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Triangle of partial sums of binomial coefficients: T(n,k) =Sum_{i=0..k} C(n,i); also dimensions of Reed-Muller codes.
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29
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1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, 1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1024, 1, 12
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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The second-left-from-middle column is A000346: T(2n+2, n) = A000346(n). - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
T(n,k) is the maximal number of regions into which n hyperplanes of co-dimension 1 divide R^k (the Cake-Without-Icing numbers) - Rob Johnson (rob(AT)whim.org), Jul 27 2008
T(n,k) gives the number of vertices within distance k (measured along the edges) of an n-dimensional unit cube, (i.e. the number of vertices on the hypercube graph Q_n whose distance from a reference vertex is <= k) [From Robert Munafo, Oct 26 2010]
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REFERENCES
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F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 376.
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flatten
Rob Johnson, Dividing Space.
Index entries for triangles and arrays related to Pascal's triangle
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FORMULA
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From partial sums across rows of Pascal triangle A007318.
T(n, 0)=1, T(n, n)=2^n, T(n, k)=T(n-1, k-1)+T(n-1, k), 0<k<n.
G.f:(1 - x*y)/((1 - y - x*y)*(1 - 2*x*y)) [From Antonio Gonzalez (gonfer00(AT)gmail.com), Sep 08 2009]
T(2n,n)=A032443(n). [From Philippe DELEHAM, Sep 16 2009]
T(n,k) = 2 T(n-1,k-1) + binomial(n-1,k) = 2 T(n-1,k) - binomial(n-1,k). [From M. F. Hasler, May 30 2010]
T(n,k) = binomial(n,n-k)*2F1(1, -k; n+1-k)(-1) - Olivier Gérard, Aug 02 2012
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EXAMPLE
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Triangle begins:
1,
1,2,
1,3,4,
1,4,7,8,
1,5,11,15,16,
1,6,16,26,31,32,
1,7,22,42,57,63,64,
1,8,29,64,99,120,127,128,
1,9,37,93,163,219,247,255,256,
1,10,46,130,256,382,466,502,511,512,
1,11,56,176,386,638,848,968,1013,1023,1024,
...
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MAPLE
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A008949 := proc(n, k) local i; add(binomial(n, i), i=0..k) end;
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MATHEMATICA
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Table[Length[Select[Subsets[n], (Length[ # ] <= k) &]], {n, 0, 12}, {k, 0, n}] // Grid [From Geoffrey Critzer, May 13 2009]
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PROG
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Contribution from M. F. Hasler, May 30 2010: (Start)
(PARI) A008949(n)=T8949(t=sqrtint(2*n-sqrtint(2*n)), n-t*(t+1)/2)
T8949(r, c)={ 2*c > r | return(sum(i=0, c, binomial(r, i))); 1<<r - sum( i=c+1, r, binomial(r, i))} (End)
(Haskell)
a008949 n k = a008949_tabl !! n !! k
a008949_row n = a008949_tabl !! n
a008949_tabl = map (scanl1 (+)) a007318_tabl
-- Reinhard Zumkeller, Nov 23 2012
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CROSSREFS
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Diagonals are given by A000079, A000225, A000295, A002663, A002664, A035038-A035042.
Columns are given by A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. [Ken Shirriff, Jun 28 2011]
Row sums sequence is A001792.
T(n, m)= A055248(n, n-m).
Cf. A110555, A007318.
Cf. A000346.
Cf. A171886. [From M. F. Hasler, May 30 2010]
Sequence in context: A039912 A163311 A210555 * A076832 A078925 A072506
Adjacent sequences: A008946 A008947 A008948 * A008950 A008951 A008952
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KEYWORD
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tabl,nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000
Corrected typo in the Maple program - R. J. Mathar, Oct 26 2010
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STATUS
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approved
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