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.

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A226975 Decimal expansion I_1(1), the modified Bessel function of the first kind.

Original entry on oeis.org

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

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Author

Horst-Holger Boltz, Jun 25 2013

Keywords

Comments

This is also the derivative of the zeroth modified Bessel function at 1.

Examples

			0.56515910399248502720769602760986330732889962162109...

References

  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 51, page 504.

Links

Crossrefs

Cf. A002454, A197036.

Programs

  • Mathematica
    RealDigits[BesselI[1, 1], 10, 110][[1]]
  • PARI
    besseli(1,1) \\ Charles R Greathouse IV, Feb 19 2014
  • SageMath
    ((1/2) * sum(1 / (4^x * factorial(x) * rising_factorial(2, x)), x, 0, oo)).n(360)
# Peter Luschny, Jan 29 2024

Formula

From Antonio Graciá Llorente, Jan 29 2024: (Start)
I_1(1) = (1/2) * Sum_{k>=0} (2*k)/(4^k*k!^2) = (1/2) * Sum_{k>=0} (2*k)/A002454(k).
Equals (1/2) * Sum_{k>=0} (4*k^2 + 4*k - 1) / (2*k)!!^2.
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k+1)/(2^k*k!).
Equals (-e) * Sum_{k>=0} (-1/2)^k * binomial(2*k,k+1)/k!
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t))*cos(t) dt. (End)

A298595 G.f.: Sum_{n>=0} a(n)*x^(2*n)/((2*n)!)^2 = 1/BesselJ(0,x).

Original entry on oeis.org

1, 1, 27, 4275, 2326275, 3260434275, 9824561849025, 56272951734424425, 560476093710119461875, 9074718916938795106861875, 226586114542199918676706160625, 8362768986063791790897266120885625, 440616849129306857329147873116900455625, 32189976281042425371050387695609814928515625
Offset: 0

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Author

Ilya Gutkovskiy, Jan 22 2018

Keywords

Examples

			1/BesselJ(0,x) = 1 + x^2/(2!)^2 + 27*x^4/(4!)^2 + 4275*x^6/(6!)^2 + 2326275*x^8/(8!)^2 + ...

Links

Crossrefs

Cf. A000275, A001818, A002454.

Programs

  • Mathematica
    nmax = 13; Table[(CoefficientList[Series[1/BesselJ[0, x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!^2)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 13; Table[(CoefficientList[Series[1/Hypergeometric0F1[1, -x^2/4], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!^2)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = ((2*n)!)^2 * [x^(2*n)] 1/BesselJ(0,x).
a(n) ~ c * Pi * 2^(4*n+3) * n^(4*n+1) / (exp(4*n) * r^(2*n+1)), where r = BesselJZero(0, 1) = A115368 = 2.40482555769... and c = 1 / BesselJ(1, r) = 1.9262348469772531439976485375138638... - Vaclav Kotesovec, May 04 2024

A384499 Number of paths from the top to bottom of a 3-dimensional triangular bipyramidal graph of height 2n, with no repeated vertices, and no upward moves.

Original entry on oeis.org

1, 15, 11475, 1093007025, 52244816853213675, 6472823166678668309527843125, 11561557982049161046080105648122197757331625, 1687343403738428640604090554388660433120115565168405371811095975
Offset: 0

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Author

Sameer Gauria, May 31 2025

Keywords

Comments

Let T(k) be a triangular grid graph as defined in Weisstein. T(k) has k*(k+1)/2 aligned triangles that are pointing the same way as T(k) and k*(k-1)/2 triangles pointing the opposite way.
The triangular bipyramidal graph TBP(n) can be constructed as follows:
- Instantiate the layers of the graph from top to bottom T(0) ... T(n-1) T(n) T(n-1) ... T(0).
- Align the (k+1)*(k+2)/2 vertices in layer T(k) with the (k+1)*(k+2)/2 aligned triangles in adjacent layer T(k+1) to create a 1:1 correspondence between vertices in T(k) and aligned triangles in T(k+1)
- Connect each vertex in T(k) with the 3 vertices of the corresponding aligned triangle in the adjacent T(k+1).
Equivalently, extending the logic from Weisstein, the triangular bipyramidal graph TBP(n) is the graph on vertices (i,j,k,l) with 0 <= i,j,k <= n, -n <= l <= n, and i+j+k+|l| == n such that vertices are adjacent if the sum of absolute differences of the coordinates of two vertices is 2, and the absolute difference of the l coordinate is 0 or 1. That is, vertices are adjacent if |i1-i2| + |j1-j2| + |k1-k2| + |l1-l2| == 2 and |l1-l2| <= 1.

Examples

			a(1)=15: TBP(1) has 5 vertices A, B, C, D, E. Vertex A (top) is connected to vertices B, C, D. B, C, D are connected to each other. B, C, D are connected to E (bottom). The only valid paths are: ABE, ACE, ADE, ABCE, ABDE, ACBE, ACDE, ADBE, ADCE, ABCDE, ABDCE, ACBDE, ACDBE, ADBCE, ADCBE. For instance, path ABCADE is not valid because of upward move (CA) and repeated vertex (A).

Links

Crossrefs

Cf. A002454 (2-dimensional version).
Previous Showing 11-13 of 13 results.

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