%I A086645
%S A086645 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,
%T A086645 66,1,1,91,1001,3003,3003,1001,91,1,1,120,1820,8008,12870,8008,1820,120,
%U A086645 1,1,153,3060,18564,43758,43758,18564,3060,153,1,1,190,4845,38760
%N A086645 Triangle read by rows: T(n; k) = Binomial(2*n; 2*k).
%C A086645 Elements have the same parity as those of Pascal's triangle.
%C A086645 Coefficients of polynomials 1/2[(1+x^(1/2))^(2n) + (1-x^(1/2))^(2n)].
%C A086645 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
%C A086645 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
%C A086645 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
%C A086645 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
%C A086645 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: 
               (Start)
%C A086645 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 A086645 The Hilbert transform of this triangle is A142992 (see A145905 for the 
               definition of this term). (End)
%C A086645 Diagonal sums : A108479 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 08 2009]
%D A086645 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 224.
%H A086645 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> [From Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008]
%F A086645 T(n; k) = (2*n)!/((2*(n-k))!*(2*k)!) row sums = A081294. COLUMNS :A000012, 
               A000384
%F A086645 Sum_{k>=0} T(n, k)*A000364(k) = A000795(n) = (2^n)*A005647(n).
%F A086645 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
%F A086645 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]
%F A086645 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]
%e A086645 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 23 2008: 
               (Start)
%e A086645 The triangle begins
%e A086645 n\k|..0.....1.....2.....3.....4.....5.....6
%e A086645 ===========================================
%e A086645 0..|..1
%e A086645 1..|..1.....1
%e A086645 2..|..1.....6.....1
%e A086645 3..|..1....15....15.....1
%e A086645 4..|..1....28....70....28.....1
%e A086645 5..|..1....45...210...210....45.....1
%e A086645 6..|..1....66...495...924...495....66.....1
%e A086645 ...
%e A086645 (End)
%p A086645 T:=(n,k)->binomial(2*n,2*k): seq(seq(T(n,k),k=0..n),n=0..12);
%o A086645 (PARI) T(n,k)=binomial(2*n,2*k)
%o A086645 (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 */
%Y A086645 Cf. A000012 A000384 A081294.
%Y A086645 Cf. A000384.
%Y A086645 A008459, A108558, A127674, A142992. [From Peter Bala (pbala(AT)toucansurf.com), 
               Oct 23 2008]
%Y A086645 Sequence in context: A082105 A143210 A152238 this_sequence A154980 A166344 
               A146766
%Y A086645 Adjacent sequences: A086642 A086643 A086644 this_sequence A086646 A086647 
               A086648
%K A086645 easy,nonn,tabl
%O A086645 0,5
%A A086645 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 26 2003
%E A086645 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2004