%I #96 Feb 10 2024 15:08:22
%S 1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,3,3,4,1,1,1,3,6,4,5,1,1,1,4,6,10,5,6,
%T 1,1,1,4,10,10,15,6,7,1,1,1,5,10,20,15,21,7,8,1,1,1,5,15,20,35,21,28,
%U 8,9,1,1,1,6,15,35,35,56,28,36,9,10,1,1,1,6,21,35,70,56,84,36,45,10,11,1,1
%N Triangle T(n, k) = binomial(floor((n+k)/2), k), n>=0, n >= k >= 0.
%C Row sums are Fibonacci(n+2). Diagonal sums are A016116. - _Paul Barry_, Jul 07 2004
%C Riordan array (1/(1-x), x/(1-x^2)). Matrix inverse is A106180. - _Paul Barry_, Apr 24 2005
%C As an infinite lower triangular matrix * [1,2,3,...] = A055244. - _Gary W. Adamson_, Dec 23 2008
%C From _Emeric Deutsch_, Jun 18 2010: (Start)
%C T(n,k) is the number of alternating parity increasing subsequences of {1,2,...,n} of size k, starting with an odd number (Terquem's problem, see the Riordan reference, p. 17). Example: T(8,5)=6 because we have 12345, 12347, 12367, 12567, 14567, and 34567.
%C T(n,k) is the number of alternating parity increasing subsequences of {1,2,...,n,n+1} of size k, starting with an even number. Example: T(7,4)=5 because we have 2345, 2347, 2367, 2567, and 4567. (End)
%C From _L. Edson Jeffery_, Mar 01 2011: (Start)
%C This triangle can be constructed as follows. Interlace two copies of the table of binomial coefficients to get the preliminary table
%C 1
%C 1
%C 1 1
%C 1 1
%C 1 2 1
%C 1 2 1
%C 1 3 3 1
%C 1 3 3 1
%C ...,
%C then shift each entire r-th column up r rows, r=0,1,2,.... Also, a signed version of this sequence (A187660 in tabular form) begins with
%C 1;
%C 1, -1;
%C 1, -1, -1;
%C 1, -2, -1, 1;
%C 1, -2, -3, 1, 1; ...
%C (compare with A066170, A130777). Let T(N,k) denote the k-th entry in row N of the signed table. Then, for N>1, row N gives the coefficients of the characteristic function p_N(x)=Sum[k=0..N, T(N,k)x^(N-k)]=0 of the N X N matrix U_N=[(0 ... 0 1);(0 ... 0 1 1);...;(0 1 ... 1);(1 ... 1)]. Now let Q_r(t) be a polynomial with recurrence relation Q_r(t)=t*Q_(r-1)(t)-Q_(r-2)(t) (r>1), with Q_0(t)=1 and Q_1(t)=t. Then p_N(x)=0 has solutions Q_(N-1)(phi_j), where phi_j=2*(-1)^(j-1)*cos(j*Pi/(2*N+1)), j=1,2,...,N.
%C For example, row N=3 is {1,-2,-1,1}, giving the coefficients of the characteristic function p_3(x)=x^3-2*x^2-x+1=0 for the 3 X 3 matrix U_3=[(0 0 1);(0 1 1);(1 1 1)], with eigenvalues Q_2(phi_j)=[2*(-1)^(j-1)*cos(j*Pi/7)]^2-1, j=1,2,3. (End)
%C Given the signed polynomials (+--++--,...) of the triangle, the largest root of the n-th row polynomial is the longest (2n+1) regular polygon diagonal length, with edge = 1. Example: the largest root to x^3 - 2x^2 - x + 1 = 0 is 2.24697...; the longest heptagon diagonal, sin(3*Pi/7)/sin(Pi/7). - _Gary W. Adamson_, Sep 06 2011
%C Given the signed polynomials from _Gary W. Adamson_'s comment, the largest root of the n-th polynomial also equals the length from the center to a corner (vertex) of a regular 2*(2*n+1)-sided polygon with side (edge) length = 1. - _L. Edson Jeffery_, Jan 01 2012
%C Put f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1). Then f(x,n) = u(x,n)/v(x,n), where u(x,n) and v(x,n) are polynomials. The flattened triangles of coefficients of u and v are both essentially A046854, as indicated by the Mathematica program headed "Polynomials". - _Clark Kimberling_, Oct 12 2014
%C From _Jeremy Dover_, Jun 07 2016: (Start)
%C T(n,k) is the number of binary strings of length n+1 starting with 0 that have exactly k pairs of consecutive 0's and no pairs of consecutive 1's.
%C T(n,k) is the number of binary strings of length n+2 starting with 1 that have exactly k pairs of consecutive 0's and no pairs of consecutive 1's. (End)
%D J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, 1978. [_Emeric Deutsch_, Jun 18 2010]
%H Nathaniel Johnston, <a href="/A046854/b046854.txt">Rows 0..100, flattened</a>
%H Robert G. Donnelly, Molly W. Dunkum, and Rachel McCoy, <a href="https://arxiv.org/abs/2303.05949">Olry Terquem's forgotten problem</a>, arXiv:2303.05949 [math.HO], 2023.
%H Jeremy M. Dover, <a href="http://arxiv.org/abs/1609.00980">Some Notes on Pairs in Binary Strings</a>, arXiv:1609.00980 [math.CO], 2016.
%H Dominique Foata and Guo-Niu Han, <a href="https://arxiv.org/abs/1304.2486">Multivariable Tangent and Secant q-derivative Polynomials</a>, arXiv:1304.2486 [math.CO], 2013; see Fig. 10.1.
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F T(n,k) = binomial(floor((n+k)/2), k).
%F G.f.: (1+x)/(1-x*y-x^2). - _Ralf Stephan_, Feb 13 2005
%F Triangle = A097806 * A049310, as infinite lower triangular matrices. - _Gary W. Adamson_, Oct 28 2007
%F T(n,k) = A065941(n,n-k) = abs(A130777(n,k)) = abs(A066170(n,k)) = abs(A187660(n,k)). - _Johannes W. Meijer_, Aug 08 2011
%F For n > 1: T(n, k) = T(n-1, k-1) + T(n-2, k), 0 < k < n-1. - _Reinhard Zumkeller_, Apr 24 2013
%F T(n,k) = A168561(n,k) + A168561(n-1,k). - _R. J. Mathar_, Feb 10 2024
%e Triangle begins:
%e 1;
%e 1 1;
%e 1 1 1;
%e 1 2 1 1;
%e 1 2 3 1 1;
%e 1 3 3 4 1 1; ...
%p A046854:= proc(n,k): binomial(floor(n/2+k/2), k) end: seq(seq(A046854(n,k),k=0..n),n=0..16); # _Nathaniel Johnston_, Jun 30 2011
%t Table[Binomial[Floor[(n+k)/2], k], {n,0,16}, {k,0,n}]//Flatten
%t (* next program: Polynomials *)
%t z = 12; f[x_, n_] := x + 1/f[x, n - 1]; f[x_, 1] = 1;
%t t = Table[Factor[f[x, n]], {n, 1, z}]
%t u = Flatten[CoefficientList[Numerator[t], x]] (* this sequence *)
%t v = Flatten[CoefficientList[Denominator[t], x]]
%t (* _Clark Kimberling_, Oct 13 2014 *)
%o (Haskell)
%o a046854 n k = a046854_tabl !! n !! k
%o a046854_row n = a046854_tabl !! n
%o a046854_tabl = [1] : f [1] [1,1] where
%o f us vs = vs : f vs (zipWith (+) (us ++ [0,0]) ([0] ++ vs))
%o -- _Reinhard Zumkeller_, Apr 24 2013
%o (PARI) T(n,k) = binomial((n+k)\2, k); \\ _G. C. Greubel_, Jul 13 2019
%o (Magma) [Binomial(Floor((n+k)/2), k): k in [0..n], n in [0..16]]; // _G. C. Greubel_, Jul 13 2019
%o (Sage) [[binomial(floor((n+k)/2), k) for k in (0..n)] for n in (0..16)] # _G. C. Greubel_, Jul 13 2019
%o (GAP) Flat(List([0..16], n-> List([0..n], k-> Binomial(Int((n+k)/2), k) ))); # _G. C. Greubel_, Jul 13 2019
%Y Reflected version of A065941, which is considered the main entry. A deficient version is in A030111.
%Y Cf. A066170, A097806, A049310, A187660.
%Y Cf. A055244. - _Gary W. Adamson_, Dec 23 2008
%K nonn,tabl,easy
%O 0,8
%A _Wouter Meeussen_