A233472 Triangle T(n,k) giving denominator of coefficient of x^k in a polynomial p(n) defined as a determinant.

1, 2, 1, 72, 12, 12, 43200, 3600, 1440, 2160, 423360000, 21168000, 4704000, 3024000, 6048000, 67212633600000, 2240421120000, 320060160000, 120022560000, 106686720000, 266716800000, 172153600393420800000, 4098895247462400000

Offset

0,2

Comments

Numerators are (-1)^k.

The polynomials p(n) satisfy the condition integral_{x=0..1} x^k*p(n) dx = 0, 0<=k<n.

References

Alexander C. Aitken, Determinants and Matrices, Oliver & Boyd (1944) page 110.

Links

Alois P. Heinz, Rows n = 0..23, flattened

Example

For n=3, the determinant of
{1,     x, x^2, x^3},
{1,   1/2, 1/3, 1/4},
{1/2, 1/3, 1/4, 1/5},
{1/3, 1/4, 1/5, 1/6}
is the polynomial p(3) = 1/43200 - x/3600 + x^2/1440 - x^3/2160.

Maple

with(LinearAlgebra):
T:= n-> (p-> seq(1/abs(coeff(p, x, k)), k=0..n))(Determinant(
         Matrix(n+1, (i, j)->`if`(i=1, x^(j-1), 1/(i+j-2))))):
seq(T(n), n=0..6);  # Alois P. Heinz, Dec 22 2013

Mathematica

a[1, k_] := x^(k-1); a[n_, k_] := 1/(n+k-2); p[m_] := Det[Table[a[n, k], {n, 1, m+1}, {k, 1, m+1}]]; t[n_, k_] := Coefficient[p[n], x, k]; Table[t[n, k] // Denominator, {n, 0, 6}, {k, 0, n}] // Flatten

Crossrefs

Sequence in context: A309207 A104024 A339144 * A284596 A216485 A096681

Adjacent sequences: A233469 A233470 A233471 * A233473 A233474 A233475

Keyword

nonn,frac,tabl,look

Author

Jean-François Alcover, Dec 11 2013

Status

approved