cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A076743 Nonzero coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the highest power of x.

Original entry on oeis.org

1, -2, 6, -2, -24, 24, 120, -240, 24, -720, 2400, -720, 5040, -25200, 15120, -720, -40320, 282240, -282240, 40320, 362880, -3386880, 5080320, -1451520, 40320, -3628800, 43545600, -91445760, 43545600, -3628800, 39916800, -598752000, 1676505600
Offset: 0

Views

Author

Mohammad K. Azarian, Nov 11 2002

Keywords

Comments

Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.
The unsigned sequence 1,2,6,2,24,24,120,240,24,720,... is n-th derivative of 1/(1-x^2). For 0<=k<=n, let a(n,k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1-x^2). If n+k is even, a(n,k)=n!*binomial(n+1,k); if n+k is odd, a(n,k)=0. The nonzero coefficients of the numerators starting with the highest power of x are 1; 2; 6,2; 24,24; ... In fact this is the (n-1)-st derivative of arctanh(x). - Rostislav Kollman (kollman(AT)dynasig.cz), Jan 04 2005

Examples

			The nonzero coefficients of the numerators starting with the highest power of x are: 1; -2; 6,-2; -24,24; ...

Crossrefs

Cf. A076256, A076257, A076741.

Programs

  • Mathematica
    a[n_, k_] := Coefficient[Expand[Together[(1+x^2)^(n+1)*D[1/(1+x^2), {x, n}]]], x, k]; Select[Flatten[Table[a[n, k], {n, 0, 10}, {k, n, 0, -1}]], #!=0&]

Formula

For 0<=k<=n, let a(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2). If n+k is even, a(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k); if n+k is odd, a(n, k)=0.

Extensions

Edited by Dean Hickerson, Nov 28 2002

A076257 Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the coefficient of the highest power of x.

Original entry on oeis.org

1, -2, 0, 6, 0, -2, -24, 0, 24, 0, 120, 0, -240, 0, 24, -720, 0, 2400, 0, -720, 0, 5040, 0, -25200, 0, 15120, 0, -720, -40320, 0, 282240, 0, -282240, 0, 40320, 0, 362880, 0, -3386880, 0, 5080320, 0, -1451520, 0, 40320, -3628800, 0, 43545600, 0, -91445760, 0, 43545600, 0
Offset: 0

Views

Author

Mohammad K. Azarian, Nov 05 2002

Keywords

Comments

Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.

Examples

			The coefficients of the numerators starting with the coefficient of the highest power of x are 1; -2,0; 6,0,-2; -24,0,24,0; ...

Crossrefs

Cf. A076256, A076741, A076743.

Programs

  • Mathematica
    a[n_, k_] := Coefficient[Expand[Together[(1+x^2)^(n+1)*D[1/(1+x^2), {x, n}]]], x, k]; Flatten[Table[a[n, k], {n, 0, 10}, {k, n, 0, -1}]]

Formula

For 0<=k<=n, let a(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2). If n+k is even, a(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k); if n+k is odd, a(n, k)=0.

Extensions

Edited by Dean Hickerson, Nov 28 2002

A076741 Nonzero coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the constant term.

Original entry on oeis.org

1, -2, -2, 6, 24, -24, 24, -240, 120, -720, 2400, -720, -720, 15120, -25200, 5040, 40320, -282240, 282240, -40320, 40320, -1451520, 5080320, -3386880, 362880, -3628800, 43545600, -91445760, 43545600, -3628800, -3628800, 199584000, -1197504000
Offset: 0

Views

Author

Mohammad K. Azarian, Nov 11 2002

Keywords

Comments

Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.

Examples

			The nonzero coefficients of the numerators starting with the constant term are: 1; -2; -2,6; 24,-24; ...

References

  • Roland Zumkeller, Formal global optimization with Taylor models, IJCAR (Ulrich Furbach and Natara jan Shankar, eds.), Lecture Notes in Computer Science, vol. 4130, Springer, 2006, pp. 408-422.

Links

Crossrefs

Cf. A076256, A076257, A076743.

Programs

  • Mathematica
    a[n_, k_] := Coefficient[Expand[Together[(1+x^2)^(n+1)*D[1/(1+x^2), {x, n}]]], x, k]; Select[Flatten[Table[a[n, k], {n, 0, 10}, {k, 0, n}]], #!=0&]

Formula

For 0<=k<=n, let a(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2). If n+k is even, a(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k); if n+k is odd, a(n, k)=0.

Extensions

Edited by Dean Hickerson, Nov 28 2002
Showing 1-3 of 3 results.

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