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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). 19
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

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.

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)

REFERENCES

A. Bj├Ârner 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.

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.

LINKS

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

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.

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

FORMULA

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)).

EXAMPLE

1;

1, 1;

1, 3, 1;

1, 5, 7, 1;

1, 7, 17, 15, 1;

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

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. A007318.

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 A047969 A047812

Adjacent sequences:  A119255 A119256 A119257 * A119259 A119260 A119261

KEYWORD

nonn,tabl

AUTHOR

Reinhard Zumkeller, May 11 2006

STATUS

approved

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Last modified April 19 17:29 EDT 2014. Contains 240767 sequences.