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A233472 Triangle T(n,k) giving denominator of coefficient of x^k in a polynomial p(n) defined as a determinant. 1
1, 2, 1, 72, 12, 12, 43200, 3600, 1440, 2160, 423360000, 21168000, 4704000, 3024000, 6048000, 67212633600000, 2240421120000, 320060160000, 120022560000, 106686720000, 266716800000, 172153600393420800000, 4098895247462400000 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 26 20:51 EDT 2022. Contains 354092 sequences. (Running on oeis4.)