

A119258


Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0<k<n: T(n,k) = 2*T(n1,k1)+T(n1,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
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OFFSET

0,5


COMMENTS

From Richard M. Green, Jul 26 2011: (Start)
T(n,nk) is the (k1)st Betti number of the subcomplex of the ndimensional half cube obtained by deleting the interiors of all halfcube shaped faces of dimension at least k.
T(n,nk) is the (k2)nd Betti number of the complement of the kequal real hyperplane arrangement in R^n.
T(n,nk) 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,nk) is the number of nodes used by the KronrodPattersonSmolyak cubature formula in numerical analysis. (End)


LINKS

Reinhard Zumkeller, Rows n=0..120 of triangle, flattened
A. Björner and V. Welker, The homology of "kequal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277313.
R. M. Green, Homology representations arising from the half cube, Adv. Math., 222 (2009), 216239.
R. M. Green, Homology representations arising from the half cube, II, J. Combin. Theory Ser. A, 117 (2010), 10371048.
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), 7193.
M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Q., 48 (2010), 290297.
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,n1) = T(2*n1,n) for n>0;
central terms give A119259; row sums give A007051;
T(n,0) = T(n,n) = 1;
T(n,1) = A005408(n1) for n>0;
T(n,2) = A056220(n1) for n>1;
T(n,n4) = A027608(n4) for n>3;
T(n,n3) = A055580(n3) for n>2;
T(n,n2) = A000337(n1) for n>1;
T(n,n1) = A000225(n) for n>0.T(n,k)=[k<=n]*(1)^k*sum{i=0..k, (1)^i*C(kn,ki)*C(n,i)};  Paul Barry, Sep 28 2007
T(n,k)=[k<=n] sum{i=nk..n, (1)^{nki}*2^{ni}*C(n,i)}.
T(n,k)=[k<=n] sum_{i=nk..n, C(n,i)*C(i1,nk1)}.
G.f. for T(n,nk): x^k/(((12x)^k)*(1x)).
T(n,k) = R(n,k,2) where R(n, k, m) = (1m)^(n+k)m^(k+1)*pochhammer(nk,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) > (1m)^(n+k)m^(k+1)*pochhammer(nk, 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



