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 1-5 of 5 results.

A093986 a(1) = 1, a(2) = 1, a(n+1) = 2n*a(n) - a(n-1). Symmetrically a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).

Original entry on oeis.org

1, 1, 3, 17, 133, 1313, 15623, 217409, 3462921, 62115169, 1238840459, 27192374929, 651378157837, 16908639728833, 472790534249487, 14166807387755777, 452865045873935377, 15383244752326047041, 553343946037863758099, 21011686704686496760721
Offset: 1

Views

Author

Amarnath Murthy, May 22 2004

Keywords

Links

Crossrefs

Cf. A093985.

Programs

  • Maple
    a[1]:=1:a[2]:=1:for n from 3 to 21 do a[n]:=2*(n-1)*a[n-1]-a[n-2] od: seq(a[n],n=1..21); # Emeric Deutsch, Apr 17 2005
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, 1, a(n-1)*(2*n-2)-a(n-2)) end:
seq(a(n), n=1..25);  # Alois P. Heinz, May 21 2013
  • Mathematica
    Table[DifferenceRoot[Function[{a, n}, {a[n] - 2*(n + 1)*a[n + 1] + a[n + 2] == 0, a[0] == 1, a[1] == 1}]][n], {n, 1, 20}]
Table[FullSimplify[((BesselY[0, 1] - BesselY[1, 1]) BesselJ[n, 1] + (BesselJ[1, 1] - BesselJ[0, 1]) BesselY[n, 1]) Pi/2], {n, 1, 20}] (* Vladimir Reshetnikov, May 21 2013 *)
RecurrenceTable[{a[1]==a[2]==1,a[n+1]==2n a[n]-a[n-1]},a,{n,20}] (* Harvey P. Dale, Jan 17 2021 *)

Formula

a(n) = ((Y_0(1) - Y_1(1))*J_n(1) + (J_1(1)-J_0(1))*Y_n(1))*Pi/2, where J_n(x) and Y_n(x) are the Bessel function of the first and second kind. - Vladimir Reshetnikov, May 21 2013
a(n) ~ sqrt(Pi/2)*(BesselJ(0,1)-BesselJ(1,1)) * 2^n*n^(n-1/2)*exp(-n). - Vaclav Kotesovec, Aug 13 2013
a(-n) = (-1)^n * a(n). - Michael Somos, May 10 2014
0 = a(n)*(a(n+2)) + a(n+1)*(-a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(a(n+2)). - Michael Somos, May 10 2014

Extensions

More terms from Emeric Deutsch, Apr 17 2005

A305466 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..floor(n/2)} ((n-j)!/j!)*binomial(n-j,j)*k^(n-2*j)*(-1)^j.

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 2, 1, 1, 4, 17, 40, 7, 0, 1, 5, 31, 150, 313, 33, -1, 1, 6, 49, 368, 1783, 3090, 191, 0, 1, 7, 71, 730, 5857, 26595, 36767, 1304, 1, 1, 8, 97, 1272, 14551, 116772, 476927, 511648, 10241, 0, 1, 9, 127, 2030, 30457, 363045, 2796671, 9988872, 8149601, 90865, -1
Offset: 0

Views

Author

Seiichi Manyama, Jun 02 2018

Keywords

Examples

			Square array begins:
    1,  1,   1,    1,    1,     1, ...
    0,  1,   2,    3,    4,     5, ...
   -1,  1,   7,   17,   31,    49, ...
    0,  2,  40,  150,  368,   730, ...
    1,  7, 313, 1783, 5857, 14551, ...

Links

Crossrefs

Columns k=0-3 give A056594, A058797, A093985(n-1), A305471.
Rows n=0-2 give A000012, A001477, A056220.
Main diagonal gives A305467.
Cf. A305401.

Formula

A(n,k) = k*n*A(n-1,k) - A(n-2,k) for n>1.

A369585 Table read by rows. T(n, k) = [z^k] h(n, 1, z) where h(n, v, z) are the modified Lommel polynomials (A369117).

Original entry on oeis.org

1, 0, 2, -1, 0, 8, 0, -8, 0, 48, 1, 0, -72, 0, 384, 0, 18, 0, -768, 0, 3840, -1, 0, 288, 0, -9600, 0, 46080, 0, -32, 0, 4800, 0, -138240, 0, 645120, 1, 0, -800, 0, 86400, 0, -2257920, 0, 10321920, 0, 50, 0, -19200, 0, 1693440, 0, -41287680, 0, 185794560
Offset: 0

Views

Author

Peter Luschny, Jan 30 2024

Keywords

Examples

			The list of coefficients starts:
  [0]  1
  [1]  0,   2
  [2] -1,   0,    8
  [3]  0,  -8,    0,   48
  [4]  1,   0,  -72,    0,   384
  [5]  0,  18,    0, -768,     0,    3840
  [6] -1,   0,  288,    0, -9600,       0,    46080
  [7]  0, -32,    0, 4800,     0, -138240,        0, 645120
  [8]  1,   0, -800,    0, 86400,       0, -2257920,      0, 10321920

Links

Crossrefs

Diagonals include: A000165 (main diagonal), A014479, A286725.
Columns include bisections of: A001105, A254371.
Cf. A093985 (row sums), A036243 (abs row sums), A369117.

Programs

  • Maple
    p := proc(n,  x) option remember; if n = -1 then 0 elif n = 0 then 1 else
2*n*z*p(n - 1, z) - p(n - 2, z) fi end:
seq(seq(coeff(p(n, z), z, k), k = 0..n), n = 0..9);
  • Mathematica
    Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, 1, 1/z]], z], {n, 0, 8}] // MatrixForm

Formula

T(n, k) = [z^k] 2*n*z*p(n-1, z) - p(n-2, z) where p(-1, z) = 0 and p(0, z) = 1.
T(n, k) = (-1)^k * [z^k] h(n, -n, z) where h(n, v, z) are the modified Lommel polynomials (A369117).

A093987 a(1) = 1, a(2) = 2, a(n+1) = a(n)^(2n)/ a(n-1). Symmetrically a(n) = {a(n-1)*a(n+1)}^[1/{(n-1) +(n+1)}].

Original entry on oeis.org

1, 2, 16, 8388608, 1532495540865888858358347027150309183618739122183602176
Offset: 1

Views

Author

Amarnath Murthy, May 22 2004

Keywords

Comments

The next term has 535 digits. - Harvey P. Dale, Jan 31 2020

Examples

			a(5) = 8388608^8/16 = 1.53249554e+54.
a(6) =~ 8.5173068734e+534.

Crossrefs

Cf. A093985, A093986.

Programs

  • Mathematica
    a[1] = 1; a[2] = 2; a[n_] := a[n - 1]^(2n - 2)/a[n - 2]; Table[ a[n], {n, 6}] (* Robert G. Wilson v, May 24 2004 *)
RecurrenceTable[{a[1]==1,a[2]==2,a[n+1]==a[n]^(2n)/a[n-1]},a,{n,5}] (* Harvey P. Dale, Jan 31 2020 *)

Extensions

Edited and extended by Robert G. Wilson v, May 24 2004

A369737 a(n) = b(n, 1/2) where b(n, x) = (Pi/4)*(Y(0, x)*J(n, x) - J(0, x)*Y(n, x)) and Y, J are Bessel functions.

Original entry on oeis.org

0, 1, 4, 31, 368, 5857, 116772, 2796671, 78190016, 2499283841, 89896028260, 3593341846559, 158017145220336, 7581229628729569, 394065923548717252, 22060110489099436543, 1323212563422417475328, 84663543948545618984449, 5755797775937679673467204, 414332776323564390870654239
Offset: 0

Views

Author

Peter Luschny, Jan 30 2024

Keywords

Crossrefs

Cf. A093985, A036243.

Programs

  • Maple
    b := (n, x) -> (Pi/4)*(BesselY(0, x)*BesselJ(n, x)-BesselJ(0, x)*BesselY(n, x)):
a := n -> simplify(b(n, 1/2)): seq(a(n), n = 0..19);
  • Mathematica
    a = (Pi/4)*(BesselY[0, 1/2] * BesselJ[n, 1/2] - BesselJ[0, 1/2] * BesselY[n, 1/2]); Table[Round[a], {n, 0, 19}]
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.