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A108299
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Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
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53
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1, 1, -1, 1, -1, -1, 1, -1, -2, 1, 1, -1, -3, 2, 1, 1, -1, -4, 3, 3, -1, 1, -1, -5, 4, 6, -3, -1, 1, -1, -6, 5, 10, -6, -4, 1, 1, -1, -7, 6, 15, -10, -10, 4, 1, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1, 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1, 1, -1, -11, 10, 45, -36, -84, 56, 70
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| Matrix inverse of A124645 .
Let L(n,x) = Sum(T(n,k)*x^(n-k): 0<=k<=n) and Pi=3.14...:
L(n,x) = Prod(x - 2*cos((2*k-1)*Pi/(2*n+1)): 1<=k<=n);
Sum(T(n,k): 0<=k<=n) = L(n,1) = A010892(n+1);
Sum(abs(T(n,k)): 0<=k<=n) = A000045(n+2);
abs(T(n,k))=A065941(n,k), T(n,k)=A065941(n,k)*A087960(k);
T(2*n,k) + T(2*n+1,k+1) = 0 for 0<=k<=2*n;
T(n,0)=A000012(n)=1; T(n,1)=-1 for n>0;
T(n,2)=-(n-1) for n>1; T(n,3)=A000027(n)=n for n>2;
T(n,4)=A000217(n-3) for n>3; T(n,5)=-A000217(n-4) for n>4;
T(n,6)=-A000292(n-5) for n>5; T(n,7)=A000292(n-6) for n>6;
T(n,n-3)=A058187(n-3)*(-1)^[n/2] for n>2;
T(n,n-2)=A008805(n-2)*(-1)^[(n+1)/2] for n>1;
T(n,n-1)=A008619(n-1)*(-1)^[n/2] for n>0;
T(n,n) = L(n,0) = (-1)^[(n+1)/2];
L(n,1) = A010892(n+1); L(n,-1) = A061347(n+2);
L(n,2) = 1; L(n,-2) = A005408(n)*(-1)^n;
L(n,3) = A001519(n); L(n,-3) = A002878(n)*(-1)^n;
L(n,4) = A001835(n+1); L(n,-4) = A001834(n)*(-1)^n;
L(n,5) = A004253(n); L(n,-5) = A030221(n)*(-1)^n;
L(n,6) = A001653(n); L(n,-6) = A002315(n)*(-1)^n;
L(n,7) = A049685(n); L(n,-7) = A033890(n)*(-1)^n;
L(n,8) = A070997(n); L(n,-8) = A057080(n)*(-1)^n;
L(n,9) = A070998(n); L(n,-9) = A057081(n)*(-1)^n;
L(n,10) = A072256(n+1); L(n,-10) = A054320(n)*(-1)^n;
L(n,11) = A078922(n+1); L(n,-11) = A097783(n)*(-1)^n;
L(n,12) = A077417(n); L(n,-12) = A077416(n)*(-1)^n;
L(n,13) = A085260(n);
L(n,14) = A001570(n); L(n,-14) = A028230(n)*(-1)^n;
L(n,n) = A108366(n); L(n,-n) = A108367(n).
Row n of the matrix inverse (A124645) has g.f.: x^[n/2]*(1-x)^(n-[n/2]). - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 12 2005
Contributed by L. Edson Jeffery, Mar 12 2011: (Start)
For odd N=2*n+1, with n>2, T(n,k) (0<=k<=n) gives the k-th coefficient in the characteristic function p_N(x)=0, of degree n in x, for the n X n tridiagonal unit-primitive "generating" matrix G_N (see [Jeffery]) of the form
G_N=A_{N,1}=
(0 1 0 ... 0)
(1 0 1 0 ... 0)
(0 1 0 1 0 ... 0)
...
(0 ... 0 1 0 1)
(0 ... 0 1 1),
with solutions phi_j=2*cos((2*j-1)*Pi/N), j=1,2,...,n. For example, for n=3,
G_7=A_{7,1}=
(0 1 0)
(1 0 1)
(0 1 1).
We have {T(3,k)}=(1,-1,-2,1), so the characteristic function of G_7 is p(x)=x^3-x^2-2*x+1=0, with solutions phi_j=2*cos((2*j-1)*Pi/7), j=1,2,3. (End)
The triangle sums, see A180662 for their definitions, link A108299 with several sequences, see the crossrefs. [Johannes W. Meijer, Aug 08 2011]
The roots to the polynomials are chaotic using iterates of the operation (x^2 - 2), with cycle lengths L and initial seeds returning to the same term or (-1)* the seed. Periodic cycle lengths L are shown in A003558 such that for the polynomial represented by row r, the cycle length L is A003558(r-1). The matrices corresponding to the rows as characteristic polynomials are likewise chaotic [Cf. Kappraff, et al, 2005] with the same cycle lengths but substituting 2*I for the "2" in (x^2 - 2), where I = the Identity matrix. For example, the roots to x^3 - x^2 - 2x + 1 = 0 are 1.801937..., -1.246979..., and .445041...With 1.801937...as the initial seed and using (x^2 - 2), we obtain the 3-period trajectory of 8.801937...-> 1.246979...-> -.445041...(returning to -1.801937...). We note that A003558(2) = 3. The corresponding matrix M is: [0,1,0; 1,0,1; 0,1,1,]. Using seed M with (x^2 - 2*I), we obtain the 3-period with the cycle completed at (-1)*M. - Gary W. Adamson, Feb 07 2012
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REFERENCES
| Friedrich L. Bauer, 'De Moivre und Lagrange: Cosinus eines rationalen Vielfachen von Pi', Informatik Spektrum 28 (Springer, 2005).
Jay Kappraff, S. Jablan, G. Adamson, & R. Sazdonovich: "Golden Fields, Generalized Fibonacci Sequences, & Chaotic Matrices"; FORMA, Vol 19, No 4, (2005).
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LINKS
| Henry W. Gould, A Variant of Pascal's Triangle , The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271.
L. E. Jeffery, Unit-primitive matrices.
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FORMULA
| T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2]
T(n+1, k) = if sign(T(n, k-1))=sign(T(n, k)) then T(n, k-1)+T(n, k) else -T(n, k-1) for 0<k<n, T(n, 0) = 1, T(n, n) = (-1)^[(n+1)/2].
G.f.: A(x, y) = (1 - x*y)/(1 - x + x^2*y^2). - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 12 2005
The generating polynomial (in z) of row n>=0 is (u^(2*n+1) + v^(2*n+1))/(u + v), where u and v are defined by u^2 + v^2 = 1 and u*v = z. [Emeric Deutsch, Jun 16 2011]
From Johannes W. Meijer, Aug 08 2011: (Start)
abs(T(n,k)) = A065941(n,k) = abs(A187660(n,n-k))
T(n,n-k) = A130777(n,k); abs(T(n,n-k)) = A046854(n,k) = abs(A066170(n,k)) (End)
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EXAMPLE
| Triangle begins :
1 ;
1, -1 ;
1, -1, -1 ;
1, -1, -2, 1 ;
1, -1, -3, 2, 1 ;
1, -1, -4, 3, 3, -1 ;
1, -1, -5, 4, 6, -3, -1 ;
1, -1, -6, 5, 10, -6, -4, 1 ;
1, -1, -7, 6, 15, -10, -10, 4, 1 ;
1, -1, -8, 7, 21, -15, -20, 10, 5, -1 ;
1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1 ;
1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1 ;...
Matrix inverse (A124645) begins:
1;
1,-1;
0,1,-1;
0,1,-2,1;
0,0,1,-2,1;
0,0,1,-3,3,-1;
0,0,0,1,-3,3,-1; ...
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MAPLE
| A108299 := proc(n, k): binomial(n-floor((k+1)/2), floor(k/2))*(-1)^floor((k+1)/2) end: seq(seq(A108299 (n, k), k=0..n), n=0..11); [Johannes W. Meijer, Aug 08 2011]
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PROG
| (PARI) {T(n, k)=polcoeff(polcoeff((1-x*y)/(1-x+x^2*y^2+x^2*O(x^n)), n, x)+y*O(y^k), k, y)} (Hanna)
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CROSSREFS
| Cf. A049310, A039961, A124645.
Triangle sums (see the comments): A193884 (Kn11), A154955 (Kn21), A087960 (Kn22), A000007 (Kn3), A010892 (Fi1), A134668 (Fi2), A078031 (Ca2), A193669 (Gi1), A001519 (Gi3), A193885 (Ze1), A050935 (Ze3). [Johannes W. Meijer, Aug 08 2011]
Cf. A003558
Sequence in context: A152157 A039961 A065941 * A123320 A054123 A119269
Adjacent sequences: A108296 A108297 A108298 * A108300 A108301 A108302
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KEYWORD
| sign,tabl,changed
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
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EXTENSIONS
| Corrected and edited. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2008
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