login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008949 Triangle of partial sums of binomial coefficients: T(n,k) =Sum_{i=0..k} C(n,i); also dimensions of Reed-Muller codes. 33
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

REFERENCES

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

LINKS

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

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)) [From 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;

MATHEMATICA

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

PROG

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

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.

Sequence in context: A039912 A163311 A210555 * A076832 A078925 A072506

Adjacent sequences:  A008946 A008947 A008948 * A008950 A008951 A008952

KEYWORD

tabl,nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

Typo in the Maple program corrected by R. J. Mathar, Oct 26 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 21 02:15 EST 2014. Contains 252291 sequences.