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A086645 Triangle read by rows: T(n; k) = Binomial(2*n; 2*k). 28
1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 28, 70, 28, 1, 1, 45, 210, 210, 45, 1, 1, 66, 495, 924, 495, 66, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1, 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1, 1, 190, 4845, 38760 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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 Deléham, 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 Deléham, 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 Deléham, 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, Apr 22 2008

From Peter Bala, 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 . [Philippe Deléham, Sep 08 2009]

Coefficients of <math>\prod_{k=1}^n(\cot^2\frac{k\pi}{2n+1}-x)=\sum_0^n\frac{(-1)^k}{2n+1-2k}{2n \choose 2k}x^k </math> [David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010]

Generalized Narayana triangle for 4^n (or cosh(2x)). [Paul Barry, Sep 28 2010]

Coefficients of the matrix inverse appear to be T^(-1)(n,k) = (-1)^(n+k)*A086646(n,k). - R. J. Mathar, Mar 12 2013

Let E(y) = sum {n >= 0} y^n/(2*n)! = cosh(sqrt(y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n)! as defined in Wang and Wang. Cf. A103327. - Peter Bala, Aug 06 2013

REFERENCES

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

LINKS

Table of n, a(n) for n=0..58.

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices arXiv:0809.5123 [math.CO]

T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

FORMULA

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). - Philippe Deléham, 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 + ... . [Peter Bala, 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 . [Philippe Deléham, Sep 08 2009]

<math>\prod_{k=1}^n(\cot^2\frac{k\pi}{2n+1}-x)=\sum_0^n\frac{(-1)^k}{2n+1-2k}{2n \choose 2k}x^k </math> [David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010]

From Paul Barry, Sep 28 2010: (Start)

G.f.: 1/(1-x-xy-4*x^2y/(1-x-xy))=(1-x(1+y))/(1-2x(1+y)+x^2(1-y)^2);

E.g.f.: exp((1+y)x)*cosh(2*sqrt(y)x);

T(n,k) = sum{j=0..n, C(n,j)*C(n-j,2(k-j))*4^(k-j)}. (End)

T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)+2*T(n-2,k-1)-T(n-2,k)-T(n-2,k-2), with T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 26 2013

EXAMPLE

Contribution from Peter Bala, 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)

From Peter Bala, Aug 06 2013: (Start)

Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n)! the column generating functions begin

1st col: cosh(sqrt(y)) = 1 + y/2! + y^2/4! + y^3/6! + y^4/8! + ....

2nd col: 1/2!*y*cosh(sqrt(y)) = y/2! + 6*y^2/4! + 15*y^3/6! + 28*y^4/8! + ....

3rd col: 1/4!*y^2*cosh(sqrt(y)) = y^2/4! + 15*y^3/6! + 70*y^4/8! + 210*y^5/10! + .... (End)

MAPLE

A086645:=(n, k)->binomial(2*n, 2*k): seq(seq(A086645(n, k), k=0..n), n=0..12);

PROG

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

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

(Maxima) create_list(binomial(2*n, 2*k), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */

CROSSREFS

Cf. A000012, A000384, A081294.

Cf. A008459, A108558, A127674, A142992. [From Peter Bala, Oct 23 2008]

Cf. A103327 (binomial(2n+1, 2k+1)), A103328 (binomial(2n, 2k+1)), A091042 (binomial(2n+1, 2k)). [Wolfdieter Lang, Jan 06 2013]

Cf. A086646 (unsigned matrix inverse), A103327.

Sequence in context: A143210 A205133 A152238 * A168291 A154980 A166344

Adjacent sequences:  A086642 A086643 A086644 * A086646 A086647 A086648

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jul 26 2003

EXTENSIONS

More terms from Emeric Deutsch, May 24 2004

STATUS

approved

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Last modified April 24 02:19 EDT 2014. Contains 240947 sequences.