0,1

Expansion T(n,x)= sum(a(n,m)*(2^(m-1))*x^m,m=0..n).

This is a signed version of triangle A114525.

The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

W. Lang, Row polynomials.

a(n,0)=0 if n is odd, a(n,0)=2*(-1)^(n/2) if n is even, else a(n,m)=t(n,m)/2^(m-1) with t(n,m):=A053120(n,m) (coefficients of Chebyshev T-polynomials).

G.f. for m-th column (signed triangle): 2/(1+x^2) if m=0 else (x^m)*(1-x^2)/(1+x^2)^(m+1). Riordan type matrix ((1-x^2)/(1+x^2),x/(1-x^2)) for n,m>=1.

a(n,m) := 0 if n<m or n+m odd; a(n,0)= 2*(-1)^(n/2) (n even); else a(n,m)=((-1)^((n+m)/ 2+m))*n*binomial((n+m)/2-1,m-1)/m.

Recursion for n >= 2 and m>=2: a(n,m) = a(n-1,m-1)-a(n-2,m), a(n,m)=0 if n<m, a(2*k,1)=0, a(2*k+1,1)=(2*k+1)*(-1)^k. In addition, for column m=0: a(2*k,0)= 2*(-1)^k, a(2*k+1,0)=0, k>=0.

Row n=4: [2,0,-4,0,1] stands for the polynomial 2*y^0 - 4*y^2 + 1*y^4. With y^m replaced by 2^(m-1)*x^m this becomes T(4,x)= 1-8*x^2+8*x^4.

[2];[0,1];[ -2,0,1];[0,-3,0,1];[2,0,-4,0,1];...

Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers).

Bisection: A127677(even n triangle, without zero entries), ((-1)^(n-m))*A111125(n, m) (odd n triangle, without zero entries).

Cf. A108045.

Sequence in context: A112177 A115723 A114525 this_sequence A134979 A112248 A010872

Adjacent sequences: A127669 A127670 A127671 this_sequence A127673 A127674 A127675

sign,tabl,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007