0,5

Elements have the same parity as those of Pascal's triangle.

Coefficients of polynomials 1/2[(1+x^(1/2))^(2n) + (1-x^(1/2))^(2n)].

Number of compositions of 2n having k parts greater than 1; example : T(3, 2) = 15 because we have 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 3+3 . - Philippe DELEHAM, May 18 2005

Number of binary words of length 2n - 1 having k runs of consecutive 1's; example : T(3,2) = 15 because we have 00101, 01001, 01010, 01011, 01101, 10001, 10010, 10011, 10100, 10110, 10111, 11001, 11010, 11011, 11101 . - Philippe DELEHAM, May 18 2005

Let M_n be the n X n matrix M_n(i, j) = T(i, j-1); then for n>0 det(M_n) = A000364(n), Euler numbers; example : det([1, 1, 0, 0; 1, 6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 = A000364(4) . -Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 04 2005

Equals ConvOffsStoT transform of the hexagonal numbers, A000384: (1, 6, 15, 28, 45,...); e.g. ConvOffs transform of (1, 6, 15, 28) = (1, 28, 70, 28, 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 22 2008

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: (Start)

Let C_n be the root lattice generated as a monoid by {+-2*e_i: 1 <= i <= n; +-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(C_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A127674 for (a signed version of) the corresponding array of f-vectors for these type C_n polytopes. See A008459 for the array of h-vectors for type A_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.

The Hilbert transform of this triangle is A142992 (see A145905 for the definition of this term). (End)

Diagonal sums : A108479 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 08 2009]

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

T(n; k) = (2*n)!/((2*(n-k))!*(2*k)!) row sums = A081294. COLUMNS :A000012, A000384

Sum_{k>=0} T(n, k)*A000364(k) = A000795(n) = (2^n)*A005647(n).

Sum_{k>=0} T(n, k)*2^k = A001541(n). Sum_{k>=0} T(n, k)*3^k = 2^n*A001075(n). Sum_{k>=0} T(n, k)*4^k = A083884(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 29 2004

O.g.f.: (1 - z*(1+x))/(x^2*z^2 - 2*x*z*(1+z) + (1-z)^2) = 1 + (1 + x)*z +(1 + 6*x + x^2)*z^2 + ... . [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Sum{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A081294(n), A001541(n), A090965(n), A083884(n), A099140(n), A099141(n), A099142(n), A165224(n), A026244(n) for x = 0,1,2,3,4,5,6,7,8,9 respectively . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 08 2009]

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: (Start)

The triangle begins

n\k|..0.....1.....2.....3.....4.....5.....6

===========================================

0..|..1

1..|..1.....1

2..|..1.....6.....1

3..|..1....15....15.....1

4..|..1....28....70....28.....1

5..|..1....45...210...210....45.....1

6..|..1....66...495...924...495....66.....1

...

(End)

T:=(n, k)->binomial(2*n, 2*k): seq(seq(T(n, k), k=0..n), n=0..12);

(PARI) T(n, k)=binomial(2*n, 2*k)

(PARI) T(n, k)=sum(i=0, min(k, n-k), 4^i*C(n, 2*i)*C(n-2*i, k-i)) /* Michael Somos May 26 2005 */

Cf. A000012 A000384 A081294.

Cf. A000384.

A008459, A108558, A127674, A142992. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]

Sequence in context: A082105 A143210 A152238 this_sequence A154980 A166344 A146766

Adjacent sequences: A086642 A086643 A086644 this_sequence A086646 A086647 A086648

easy,nonn,tabl

DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 26 2003

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2004