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A008949 Triangle of partial sums of binomial coefficients: T(n,k) =Sum_{i=0..k} C(n,i); also dimensions of Reed-Muller codes. 36
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)
OFFSET

0,3

COMMENTS

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, 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). - Robert Munafo, Oct 26 2010

A triangle formed like Pascal's triangle, but with 2^n n>=0 on the right border instead of 1. - Boris Putievskiy, Aug 18 2013

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013

T(n,floor(n/2)) = A027306(n). - Reinhard Zumkeller, Nov 14 2014

Consider each "1" as an apex of two sequences: the first is the set of terms in the same row as the "1", but the rightmost term in the row repeats infinitely. Example: the row (1, 4, 7, 8) becomes (1, 4, 7, 8, 8, 8,...). The second sequence begins with the same "1" but is the diagonal going down and to the right, thus: (1, 5, 16, 42, 99, 219, 466,...). It appears that for all such sequence pairs, the binomial transform of the first, (1, 4, 7, 8, 8, 8,...) in this case; is equal to the second: (1, 5, 16, 42, 99,...). - Gary W. Adamson, Aug 19 2015

REFERENCES

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 376.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Rob Johnson, Dividing Space.

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

FORMULA

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)). - Antonio Gonzalez (gonfer00(AT)gmail.com), Sep 08 2009

T(2n,n) = A032443(n). - Philippe Deléham, 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). - 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

For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013

EXAMPLE

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,

...

MAPLE

A008949 := proc(n, k) local i; add(binomial(n, i), i=0..k) end; # Typo corrected by R. J. Mathar, Oct 26 2010

MATHEMATICA

Table[Length[Select[Subsets[n], (Length[ # ] <= k) &]], {n, 0, 12}, {k, 0, n}] // Grid (* Geoffrey Critzer, May 13 2009 *)

Flatten[Accumulate/@Table[Binomial[n, i], {n, 0, 20}, {i, 0, n}]] (* Harvey P. Dale, Aug 08 2015 *)

PROG

(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))} \\ M. F. Hasler, May 30 2010

(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

CROSSREFS

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, A000346, A171886, A228196, A228576.

Cf. A027306, A249111.

Cf. A163866.

Cf. A261363.

Sequence in context: A039912 A163311 A210555 * A076832 A078925 A072506

Adjacent sequences:  A008946 A008947 A008948 * A008950 A008951 A008952

KEYWORD

tabl,nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 23 2000

STATUS

approved

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Last modified August 31 03:30 EDT 2015. Contains 261232 sequences.