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

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A119258 Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0<k<n: T(n,k) = 2*T(n-1,k-1)+T(n-1,k). 20
1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 17, 15, 1, 1, 9, 31, 49, 31, 1, 1, 11, 49, 111, 129, 63, 1, 1, 13, 71, 209, 351, 321, 127, 1, 1, 15, 97, 351, 769, 1023, 769, 255, 1, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 511, 1, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1023, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

From Richard M. Green, Jul 26 2011: (Start)

T(n,n-k) is the (k-1)-st Betti number of the subcomplex of the n-dimensional half cube obtained by deleting the interiors of all half-cube shaped faces of dimension at least k.

T(n,n-k) is the (k-2)-nd Betti number of the complement of the k-equal real hyperplane arrangement in R^n.

T(n,n-k) gives a lower bound for the complexity of the problem of determining, given n real numbers, whether some k of them are equal.

T(n,n-k) is the number of nodes used by the Kronrod-Patterson-Smolyak cubature formula in numerical analysis. (End)

LINKS

Reinhard Zumkeller, Rows n=0..120 of triangle, flattened

A. Björner and V. Welker, The homology of "k-equal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277-313.

R. M. Green, Homology representations arising from the half cube, Adv. Math., 222 (2009), 216-239.

R. M. Green, Homology representations arising from the half cube, II, J. Combin. Theory Ser. A, 117 (2010), 1037-1048.

R. M. Green, Homology representations arising from the half cube, II, arXiv:0812.1208 [math.RT], 2008

R. M. Green and Jacob T. Harper, Morse matchings on polytopes, Arxiv preprint arXiv:1107.4993, 2011

OEIS Wiki, Sequence of the Day for November 3.

K. Petras, On the Smolyak cubature error for analytic functions, Adv. Comput. Math., 12 (2000), 71-93.

M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Q., 48 (2010), 290-297.

M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, arXiv:1107.1063 [math.CO], 2010

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

FORMULA

T(2*n,n-1) = T(2*n-1,n) for n>0;

central terms give A119259; row sums give A007051;

T(n,0) = T(n,n) = 1;

T(n,1) = A005408(n-1) for n>0;

T(n,2) = A056220(n-1) for n>1;

T(n,n-4) = A027608(n-4) for n>3;

T(n,n-3) = A055580(n-3) for n>2;

T(n,n-2) = A000337(n-1) for n>1;

T(n,n-1) = A000225(n) for n>0.T(n,k)=[k<=n]*(-1)^k*sum{i=0..k, (-1)^i*C(k-n,k-i)*C(n,i)}; - Paul Barry, Sep 28 2007

T(n,k)=[k<=n] sum{i=n-k..n, (-1)^{n-k-i}*2^{n-i}*C(n,i)}.

T(n,k)=[k<=n] sum_{i=n-k..n, C(n,i)*C(i-1,n-k-1)}.

G.f. for T(n,n-k): x^k/(((1-2x)^k)*(1-x)).

T(n,k) = R(n,k,2) where R(n, k, m) = (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k,k+1)* hyper2F1([1,n+1], [k+2], m)/(k+1)!. - Peter Luschny, Jul 25 2014

EXAMPLE

1;

1, 1;

1, 3, 1;

1, 5, 7, 1;

1, 7, 17, 15, 1;

1, 9, 31, 49, 31, 1;

MAPLE

# Case m = 2 of the more general:

A119258 := (n, k, m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1)*hypergeom([1, n+1], [k+2], m)/(k+1)!;

seq(seq(round(evalf(A119258(n, k, 2))), k=0..n), n=0..10); # Peter Luschny, Jul 25 2014

PROG

(Haskell)

a119258 n k = a119258_tabl !! n !! k

a119258_row n = a119258_tabl !! n

a119258_list = concat a119258_tabl

a119258_tabl = iterate (\row -> zipWith (+)

   ([0] ++ init row ++ [0]) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]

-- Reinhard Zumkeller, Nov 15 2011

CROSSREFS

Cf. A119259, A007318, A007051, A005408, A056220, A027608, A055580, A000337, A000225.

A145661, A119258 and A118801 are all essentially the same (see the Shattuck and Waldhauser paper). - Tamas Waldhauser, Jul 25 2011

Sequence in context: A216948 A183944 A145661 * A099608 A247285 A047969

Adjacent sequences:  A119255 A119256 A119257 * A119259 A119260 A119261

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, May 11 2006

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 September 23 01:51 EDT 2014. Contains 247086 sequences.