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 A086645 Triangle read by rows: T(n, k) = binomial(2n, 2k). 37
 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 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 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 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 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224. LINKS Indranil Ghosh, Rows 0.. 120 of triangle, flattened F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices arXiv:0809.5123 [math.CO], 2008. H. Chan, S. Cooper, P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543. T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012. T. Copeland, Skipping over Dimensions, Juggling Zeros in the Matrix, 2020. 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..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 $\prod_{k=1}^n(\cot^2\frac{k\pi}{2n+1}-x)=\sum_0^n\frac{(-1)^k}{2n+1-2k}{2n \choose 2k}x^k$. - David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010 From Paul Barry, Sep 28 2010: (Start) 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); 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); MATHEMATICA Table[Binomial[2 n, 2 k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 13 2016 *) 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 */ (MAGMA) /* As triangle: */ [[Binomial(2*n, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 14 2016 CROSSREFS Cf. A098158, A000384, A081294. Cf. A008459, A108558, A127674, A142992. [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. Cf. A034839. Sequence in context: A205133 A152238 A295985 * A168291 A154980 A166344 Adjacent sequences:  A086642 A086643 A086644 * A086646 A086647 A086648 KEYWORD nonn,tabl,easy AUTHOR Philippe Deléham, Jul 26 2003 STATUS approved

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Last modified January 15 16:05 EST 2021. Contains 340187 sequences. (Running on oeis4.)