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 1-7 of 7 results.

A085528 a(n) = (2*n+1)^(n+1).

Original entry on oeis.org

1, 9, 125, 2401, 59049, 1771561, 62748517, 2562890625, 118587876497, 6131066257801, 350277500542221, 21914624432020321, 1490116119384765625, 109418989131512359209, 8629188747598184440949, 727423121747185263828481, 65273511648264442971824673
Offset: 0

Views

Author

N. J. A. Sloane, Jul 05 2003

Keywords

Comments

a(n) is the number of polynomials of degree at most n with integer coefficients all having absolute value <= n.
a(n-1) is the number of nodes in the canonical automaton for the affine Weyl group of type D_n. - Tom Edgar, May 12 2016

References

  • Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.

Links

Crossrefs

Cf. A000169, A085527, A099753, A214406, A253299.

Programs

Formula

From Peter Bala, Aug 06 2012: (Start)
E.g.f.: d/dx{(2*x/T(2*x))^(1/2)*1/(1 - T(2*x))} = 1 + 9*x + 125*x^2/2! + ..., where T(x) is the tree function sum {n >= 1} n^(n-1)*x^n/n! of A000169.
For r = 0, 1, 2, ... the e.g.f. for the sequence (2*n+1)^(n+r) can be expressed in terms of the function U(z) = sum {n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 1, and the resulting e.g.f. is 1/z*U(z)*(1 + U(z)^2 )/(1 - U(z)^2)^3 taken at z = sqrt(2*x).
(End)
Sum_{n>=0} (-1)^n/a(n) = A253299. - Amiram Eldar, Jun 25 2021

A359282 Decimal expansion of Integral_{x = 0..1} 1/x^(x^2) dx.

Original entry on oeis.org

1, 1, 1, 9, 5, 4, 5, 1, 2, 0, 1, 3, 6, 1, 2, 7, 5, 9, 6, 6, 1, 2, 6, 7, 6, 2, 4, 7, 0, 2, 9, 8, 2, 7, 0, 3, 6, 4, 6, 0, 0, 4, 6, 9, 5, 7, 8, 7, 6, 4, 2, 7, 6, 2, 8, 9, 8, 6, 7, 4, 9, 5, 4, 6, 7, 5, 7, 0, 9, 4, 4, 0, 8, 3, 4, 4, 3, 2, 8, 3, 9, 8, 7, 5, 6, 8, 6, 2, 6, 4, 5, 3, 8, 2, 0, 1, 0, 7, 7, 3, 0, 0, 5, 9, 7, 9, 9, 4
Offset: 1

Views

Author

Peter Bala, Dec 24 2022

Keywords

Examples

			1.119545120136127596612676247029827036460046957876427628986749 ...

Links

Crossrefs

Cf. A073009, A083648, A253299, A359283, A359284, A359285, A359286.

Programs

  • Maple
    evalf(Sum(1/(2*n-1)^n, n = 1..infinity), 120);
  • Mathematica
    NIntegrate[x^(-x^2), {x, 0, 1}, WorkingPrecision -> 103] // RealDigits // First
  • PARI
    intnum(x = 0, 1, x^(-x^2))

Formula

Equals Sum_{n >= 1} 1/(2*n - 1)^n.
More generally, Integral_{x = 0..1} 1/x^(t*x^2) dx = Sum_{n >= 1} t^(n-1)/(2*n - 1)^n. See A253299 (case t = -1).

A359283 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^2) dx.

Original entry on oeis.org

4, 6, 2, 3, 0, 3, 7, 1, 1, 5, 3, 7, 3, 2, 1, 0, 7, 7, 1, 8, 2, 0, 3, 9, 6, 2, 8, 5, 8, 8, 2, 7, 7, 4, 4, 0, 9, 6, 1, 0, 2, 6, 0, 3, 7, 0, 4, 8, 4, 0, 7, 5, 6, 2, 2, 7, 0, 1, 3, 0, 0, 6, 0, 2, 5, 6, 7, 8, 2, 3, 3, 7, 7, 0, 2, 4, 0, 9, 8, 4, 4, 7, 7, 3, 4, 1, 7, 5, 4, 6, 1, 0, 5, 4, 2, 3, 3, 8, 6, 1, 8
Offset: 0

Views

Author

Peter Bala, Dec 24 2022

Keywords

Comments

For a, b nonnegative integers, the alternating divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx.

Examples

			0.46230371153732107718203962858827744096102603704840...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359284, A359285, A359286.

Programs

  • Maple
    evalf(int(1/x^(x^2), x = 1..infinity), 100);
  • Mathematica
    NIntegrate[1/x^(x^2), {x, 1, Infinity}, WorkingPrecision -> 105] // RealDigits // First

Formula

Equals Integral_{x = 1..oo} 1/(2*x - 1)^x dx.
Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(2*n + 1)^n. Compare with the alternating convergent series Sum_{n >= 1} (-1)^(n+1)/(2*n - 1)^n = Integral_{x = 0..1} x^(x^2) dx. See A253299.

A359284 Decimal expansion of Integral_{x = 0..1} 1/x^(x^3) dx.

Original entry on oeis.org

1, 0, 6, 5, 5, 1, 8, 2, 0, 5, 9, 2, 7, 6, 4, 9, 1, 7, 5, 8, 6, 3, 8, 2, 1, 4, 0, 5, 4, 8, 4, 5, 4, 7, 2, 3, 1, 5, 3, 9, 8, 0, 2, 2, 7, 9, 0, 9, 9, 8, 2, 1, 2, 4, 8, 9, 8, 9, 2, 8, 4, 5, 6, 5, 8, 7, 8, 3, 0, 3, 2, 5, 6, 8, 1, 2, 4, 5, 7, 0, 0, 0, 3, 8, 3, 0, 1, 9, 3, 5, 7, 6, 1, 2, 3, 9, 9, 4, 0, 9, 2, 8, 7, 9, 2, 7, 9, 0
Offset: 1

Views

Author

Peter Bala, Dec 24 2022

Keywords

Examples

			1.06551820592764917586382140548454723153980227909982...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359283, A359285, A359286.

Programs

  • Maple
    evalf(int(1/x^(x^3), x = 0..1), 110);
  • Mathematica
    NIntegrate[1/x^(x^3), {x, 0, 1}, WorkingPrecision -> 110] // RealDigits // First
  • PARI
    intnum(x = 0, 1, x^(-x^3))

Formula

Equals Sum_{n >= 1} 1/(3*n - 2)^n.
More generally, Integral_{x = 0..1} 1/x^(t*x^3) dx = Sum_{n >= 1} t^(n-1)/(3*n - 2)^n. See A359285 (case t = -1).

A359285 Decimal expansion of Integral_{x = 0..1} x^(x^3) dx.

Original entry on oeis.org

9, 4, 0, 3, 1, 8, 0, 8, 6, 6, 8, 1, 9, 0, 6, 9, 8, 2, 8, 9, 7, 3, 6, 5, 6, 4, 1, 7, 4, 2, 9, 7, 6, 7, 2, 5, 8, 1, 1, 7, 5, 1, 1, 0, 1, 4, 9, 3, 0, 6, 7, 3, 5, 2, 9, 2, 6, 6, 6, 5, 1, 0, 1, 8, 3, 8, 0, 3, 5, 8, 9, 2, 2, 7, 3, 1, 2, 4, 6, 1, 7, 1, 5, 4, 6, 4, 0, 0, 8, 3, 6, 6, 6, 6, 7, 3, 3, 7, 7, 1, 2, 8, 1, 9, 3, 0, 7, 2, 6, 7
Offset: 0

Views

Author

Peter Bala, Dec 24 2022

Keywords

Examples

			0.94031808668190698289736564174297672581175110149306...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359283, A359284, A359286.

Programs

  • Maple
    evalf(int(x^(x^3), x = 0..1), 110);
  • Mathematica
    NIntegrate[x^(x^3), {x, 0, 1}, WorkingPrecision -> 110] // RealDigits // First
  • PARI
    intnum(x = 0, 1, x^(x^3))

Formula

Equals Sum_{n >= 1} (-1)^(n+1)/(3*n - 2)^n.

A359286 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^3) dx.

Original entry on oeis.org

3, 5, 8, 5, 4, 2, 7, 1, 6, 0, 0, 0, 3, 3, 9, 9, 6, 5, 7, 0, 7, 0, 5, 7, 6, 0, 7, 7, 9, 1, 8, 1, 1, 3, 1, 1, 6, 8, 2, 0, 3, 6, 2, 0, 5, 7, 2, 1, 3, 0, 1, 1, 2, 7, 7, 0, 4, 0, 0, 8, 7, 6, 4, 8, 8, 1, 4, 0, 5, 6, 5, 4, 1, 2, 9, 1, 5, 9, 7, 3, 0, 1, 1, 4, 9, 3, 2, 5, 3, 6, 1, 5, 7, 6, 5, 9, 5, 6, 9, 9, 7, 4, 4, 0, 3, 6, 8, 6
Offset: 1

Views

Author

Peter Bala, Dec 24 2022

Keywords

Comments

For a, b nonnegative integers, the alternating divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx.

Examples

			0.35854271600033996570705760779181131168203620572130...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359283, A359284, A359285.

Programs

  • Maple
    evalf(int(1/x^(x^3), x = 1..infinity), 110);
  • Mathematica
    NIntegrate[1/x^(x^3), {x, 1, Infinity}, WorkingPrecision -> 110] // RealDigits // First

Formula

Equals Integral_{x = 1..oo} 1/(3*x - 2)^x dx.
Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(3*n + 2)^n. Compare with the alternating convergent series Sum_{n >= 1} (-1)^(n+1)/(3*n - 2)^n = Integral_{x = 0..1} x^(x^3) dx. See A359285.

A253300 Decimal expansion of integral_{x=0..1} x^sqrt(x) dx.

Original entry on oeis.org

6, 5, 8, 5, 8, 2, 3, 5, 4, 1, 0, 9, 0, 9, 3, 5, 6, 5, 4, 6, 9, 6, 5, 6, 8, 5, 3, 4, 0, 3, 6, 4, 4, 1, 7, 0, 1, 5, 6, 4, 0, 5, 8, 9, 2, 7, 7, 3, 3, 6, 2, 4, 6, 1, 1, 3, 3, 7, 5, 8, 6, 2, 6, 4, 2, 6, 5, 4, 6, 7, 1, 7, 8, 8, 7, 9, 8, 7, 1, 9, 5, 7, 8, 8, 8, 1, 4, 1, 6, 4, 6, 8, 5, 9, 1, 1, 3, 9, 0, 2, 9, 8, 6, 4, 6
Offset: 0

Views

Author

Jean-François Alcover, Dec 30 2014

Keywords

Examples

			0.6585823541090935654696568534036441701564...

References

  • Paul J. Nahin, Inside Interesting Integrals, Springer 2014, ISBN 978-1493912766.

Links

Crossrefs

Cf. A073009, A083648, A253299.

Programs

  • Mathematica
    NIntegrate[x^Sqrt[x], {x, 0, 1}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First
  • PARI
    intnum(x=0,1, x^sqrt(x)) \\ Michel Marcus, Dec 30 2014

Formula

Equals sum_{n >= 1} (-1)^(n + 1)*(2/(n + 1))^n.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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