A086645 - OEIS

%I #101 Oct 13 2022 15:11:37

%S 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,

%T 495,66,1,1,91,1001,3003,3003,1001,91,1,1,120,1820,8008,12870,8008,

%U 1820,120,1,1,153,3060,18564,43758,43758,18564,3060,153,1,1,190,4845,38760

%N Triangle read by rows: T(n, k) = binomial(2n, 2k).

%C Terms have the same parity as those of Pascal's triangle.

%C Coefficients of polynomials (1/2)*((1 + x^(1/2))^(2n) + (1 - x^(1/2))^(2n)).

%C 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

%C 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

%C 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

%C 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

%C From _Peter Bala_, Oct 23 2008: (Start)

%C 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.

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

%C (End)

%C Diagonal sums: A108479. - _Philippe Deléham_, Sep 08 2009

%C Coefficients of Product_{k=1..n} (cot(k*Pi/(2n+1))^2 - x) = Sum_{k=0..n} (-1)^k*binomial(2n,2k)*x^k/(2n+1-2k). - David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010

%C Generalized Narayana triangle for 4^n (or cosh(2x)). - _Paul Barry_, Sep 28 2010

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

%C 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

%C Row 6, (1,66,495,924,495,66,1), plays a role in expansions of powers of the Dedekind eta function. See the Chan link, p. 534, and A034839. - _Tom Copeland_, Dec 12 2016

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

%H Indranil Ghosh, <a href="/A086645/b086645.txt">Rows 0.. 120 of triangle, flattened</a>

%H F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, <a href="http://arxiv.org/abs/0809.5123">Root polytopes and growth series of root lattices</a> arXiv:0809.5123 [math.CO], 2008.

%H H. Chan, S. Cooper and P. Toh, <a href="http://dx.doi.org/10.1016/j.aim.2005.12.003">The 26th power of Dedekind's eta function</a> Advances in Mathematics, 207 (2006) 532-543.

%H T. Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>, 2012.

%H T. Copeland, <a href="https://tcjpn.wordpress.com/2020/07/11/skipping-over-dimensions-juggling-zeros-in-the-matrix/">Skipping over Dimensions, Juggling Zeros in the Matrix</a>, 2020.

%H E. Lucas, <a href="http://edouardlucas.free.fr/gb/pdf/th_des_fonctions.pdf">Théorie des fonctions numériques simplement périodiques</a>, Baltimore, 1878. See page 145 equation (153).

%H W. Wang and T. Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

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

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

%F 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

%F 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

%F Sum_{k=0..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

%F Product_{k=1..n} (cot(k*Pi/(2n+1))^2 - x) = Sum_{k=0..n} (-1)^k*binomial(2n,2k)*x^k/(2n+1-2k). - David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010

%F From _Paul Barry_, Sep 28 2010: (Start)

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

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

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

%F 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

%F From _Peter Bala_, Sep 22 2021: (Start)

%F n-th row polynomial R(n,x) = (1-x)^n*T(n,(1+x)/(1-x)), where T(n,x) is the n-th Chebyshev polynomial of the first kind. Cf. A008459.

%F R(n,x) = Sum_{k = 0..n} binomial(n,2*k)*(4*x)^k*(1 + x)^(n-2*k).

%F R(n,x) = n*Sum_{k = 0..n} (n+k-1)!/((n-k)!*(2*k)!)*(4*x)^k*(1-x)^(n-k) for n >= 1. (End)

%e From _Peter Bala_, Oct 23 2008: (Start)

%e The triangle begins

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

%e ===========================================

%e 0..|..1

%e 1..|..1.....1

%e 2..|..1.....6.....1

%e 3..|..1....15....15.....1

%e 4..|..1....28....70....28.....1

%e 5..|..1....45...210...210....45.....1

%e 6..|..1....66...495...924...495....66.....1

%e ...

%e (End)

%e From _Peter Bala_, Aug 06 2013: (Start)

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

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

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

%e 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)

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

%t Table[Binomial[2 n, 2 k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 13 2016 *)

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

%o (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 */

%o (Maxima) create_list(binomial(2*n,2*k),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */

%o (Magma) /* As triangle: */ [[Binomial(2*n, 2*k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Dec 14 2016

%Y Cf. A098158, A000384, A081294.

%Y Cf. A008459, A108558, A127674, A142992. - _Peter Bala_, Oct 23 2008

%Y Cf. A103327 (binomial(2n+1, 2k+1)), A103328 (binomial(2n, 2k+1)), A091042 (binomial(2n+1, 2k)). -_Wolfdieter Lang_, Jan 06 2013

%Y Cf. A086646 (unsigned matrix inverse), A103327.

%Y Cf. A034839.

%K nonn,tabl,easy

%O 0,5

%A _Philippe Deléham_, Jul 26 2003