A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -2, 0, 0, 0, 1, 0, -3, 0, -1, 0, 1, 2, 0, -3, 0, -2, 0, 1, 0, 5, 0, -2, 0, -3, 0, 1, -2, 0, 8, 0, 0, 0, -4, 0, 1, 0, -7, 0, 10, 0, 3, 0, -5, 0, 1, 2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1, 0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1, -2, 0, 24, 0, -35, 0, 0, 0, 18, 0, -8, 0, 1, 0, -11, 0, 49, 0, -42, 0, -12, 0

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Mar 13 2008

Keywords

• sign
• tabl

Comments

Polynomial coefficients of P(n,x) in increasing powers, read by rows, where P(0,x)=1, P(1,x)=x, P(2,x)=2+x^2, P(3,x)=x+x^3, P(4,x)=-2+x^4, and P(n,x) = x*P(n-1,x) - P(n-2,x) for n>=5.

The row-reversed version of A135929.

Row sums are repeating 1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1..., see A138034 and A119910.

Examples

			{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-2, 0, 0, 0, 1}, = -2+x^4
{0, -3, 0, -1, 0, 1}, = -3x-x^3+x^5
{2, 0, -3, 0, -2, 0, 1},
{0, 5, 0, -2, 0, -3, 0, 1},
{-2, 0, 8, 0, 0, 0, -4, 0, 1},
{0, -7, 0, 10, 0, 3, 0, -5, 0, 1},
{2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1},
{0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1}
		

Links

• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165

Crossrefs

Cf. A123956, A137289.

Programs

Maple

A137276 := proc(n,k) local nmk,npk; if n = 0 then 1; elif (n-k) mod 2 <> 0 then 0; else nmk := (n-k)/2 ; npk := (n+k)/2 ; (-1)^nmk*(2*k-n)/npk*binomial(npk,nmk) ; fi; end:
seq( seq(A137276(n,k),k=0..n),n=0..13) ;

Formula

T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

P(n,x) = x*P(n-1,x)-P(n-2,x), n>=5.

P(n,2*x) = -2*T(n,x)+4*x*U(n-1,x), where T(n,x) is A053120 and U(n,x) is A053117.

Extensions

Fourth row inserted by the Associate Editors of the OEIS, Aug 27 2009