A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2)* binomial(n - floor((k + 1)/2), floor(k/2)).

1, 1, -2, -1, 1, -2, -6, 2, 1, 1, -2, -11, 12, 11, -2, -1, 1, -2, -16, 22, 46, -22, -16, 2, 1, 1, -2, -21, 32, 106, -92, -106, 32, 21, -2, -1, 1, -2, -26, 42, 191, -212, -396, 212, 191, -42, -26, 2, 1, 1, -2, -31, 52, 301, -382, -1011, 792, 1011, -382, -301, 52, 31, -2, -1

Offset

1,3

Links

G. C. Greubel, Rows n = 1..101 of triangle, flattened

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

Formula

Let f(n, k) = (-1)^floor((k + 1)/2)*binomial(n - floor((k + 1)/2), floor(k/2)) then the polynomials P(n, k, x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k) for an irregular triangle of coefficients.

Example

Triangle begins with:
1;
1, -2,  -1;
1, -2,  -6,  2,  1;
1, -2, -11, 12, 11,  -2,  -1;
1, -2, -16, 22, 46, -22, -16, 2, 1;

Mathematica

f[n_, k_]:= (-1)^Floor[(k+1)/2]*Binomial[n -Floor[(k+1)/2], Floor[k/2]]; Table[CoefficientList[Sum[f[n, k]*(2*x)^k*(1-x^2)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten

Crossrefs

Cf. A038223, A122600, A122753.

Sequence in context: A278543 A113186 A206497 * A298608 A327815 A007736

Adjacent sequences: A123215 A123216 A123217 * A123219 A123220 A123221

Keyword

tabf,sign

Author

Roger L. Bagula and Gary W. Adamson, Oct 04 2006

Status

approved