0,8

G.f. for row polynomials S(n,x) (signed triangle): 1/(1-x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,x) as row polynomials with G.f. 1/(1-x*z-z^2). |a(n,m)| triangle has rows of Pascal's triangle A007318 in the even numbered diagonals (odd numbered ones have only 0's).

Row sums (unsigned triangle) A000045(n+1) (Fibonacci). Row sums (signed triangle) S(n,1) sequence = periodic(1,1,0,-1,-1,0) = A010892.

S(n,x) is the characteristic polynomial of the adjacency matrix of the n-path. - Michael Somos, Jun 24 2002

|T(n,k)|=number of compositions of n+1 into k+1 odd parts. Example: |T(7,3)|=10 because we have (1,1,3,3),(1,3,1,3),(1,3,3,1),(3,1,1,3),(3,1,3,1),(3,3,1,1), (1,1,1,5),(1,1,5,1),(1,5,1,1) and (5,1,1,1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005

D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Rows 0 to 100 of the triangle, flattened.

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Index entries for sequences related to Chebyshev polynomials.

T(n, k) := 0 if n<k or n+k odd, else ((-1)^((n+k)/2+k))*binomial((n+k)/2, k); T(n, k) = -T(n-2, k)+T(n-1, k-1), T(n, -1) := 0 =: T(-1, k), T(0, 0)=1, T(n, k)= 0 if n<k or n+k odd; G.f. k-th column: (1/(1+x^2)^(k+1))*x^k. - Michael Somos, Jun 24 2002

T(n, k)=binomial((n+k)/2, (n-k)/2)*cos(pi*(n-k)/2)*(1+(-1)^(n-k))/2; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005

Sum_{k=0..n} T(n, k)^2 = A051286(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2005

Recurrence for the (unsigned) Fibonacci polynomials: F[1]=1, F[2]=x; for n>2, F[n] = x*F[n-1]+F[n-2].

{1}; {0,1}; {-1,0,1}; {0,-2,0,1}; {1,0,-3,0,1};... E.g. fourth row {0,-2,0,1} corresponds to polynomial S(3,x)= -2*x+x^3.

Triangle of absolute values of coefficients (coefficients of Fibonacci polynomials) with exponents in increasing order begins:

[1]

[0, 1]

[1, 0, 1]

[0, 2, 0, 1]

[1, 0, 3, 0, 1]

[0, 3, 0, 4, 0, 1]

[1, 0, 6, 0, 5, 0, 1]

[0, 4, 0, 10, 0, 6, 0, 1]

[1, 0, 10, 0, 15, 0, 7, 0, 1]

[0, 5, 0, 20, 0, 21, 0, 8, 0, 1]

(PARI) T(n, k)=if(k<0|k>n|(n+k)%2, 0, (-1)^((n+k)/2+k)*binomial((n+k)/2, k))

Cf. A010892.

Reflection of A053119. Without zeros: A053112.

Sequence in context: A029407 A099544 A036414 this_sequence A036851 A036850 A113206

Adjacent sequences: A049307 A049308 A049309 this_sequence A049311 A049312 A049313

easy,nice,sign,tabl,core

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)