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A086645
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Triangle read by rows: T(n; k) = Binomial(2*n; 2*k).
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25
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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; internal format)
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OFFSET
| 0,5
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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 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]
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> [From David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010]
Generalized Narayana triangle for 4^n (or cosh(2x)). [From Paul Barry (pbarry(AT)wit.ie), Sep 28 2010]
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
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LINKS
| 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]
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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). - 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]
<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> [From David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010]
Contribution from Paul Barry (pbarry(AT)wit.ie), 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)
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EXAMPLE
| 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)
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MAPLE
| T:=(n, k)->binomial(2*n, 2*k): seq(seq(T(n, k), k=0..n), n=0..12);
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PROG
| (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 */
(Maxima) create_list(binomial(2*n, 2*k), n, 0, 12, k, 0, n); [Emanuele Munarini, Mar 11 2011]
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CROSSREFS
| Cf. A000012 A000384 A081294.
Cf. A000384.
A008459, A108558, A127674, A142992. [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]
Sequence in context: A143210 A205133 A152238 * A168291 A154980 A166344
Adjacent sequences: A086642 A086643 A086644 * A086646 A086647 A086648
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 26 2003
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2004
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