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 1-6 of 6 results.

A036242 Numerator of fraction equal to the continued fraction [0,2,4,...2n].

Original entry on oeis.org

1, 4, 25, 204, 2065, 24984, 351841, 5654440, 102131761, 2048289660, 45164504281, 1085996392404, 28281070706785, 792955976182384, 23816960356178305, 762935687373888144, 25963630331068375201, 935453627605835395380, 35573201479352813399641, 1423863512801718371381020
Offset: 1

Views

Author

Jeff Burch

Keywords

Crossrefs

Cf. A036243 (denominator), A058798.

Programs

  • Mathematica
    a[n_] := FromContinuedFraction[Range[0, 2n, 2]] // Numerator; Array[a,20] (* Jean-François Alcover, Jun 03 2019 *)
  • Sage
    def A036242(n):
    if n == 1: return 1
    return 2^(n-1)*factorial(n)*hypergeometric([1/2 - n/2, 1 - n/2], [2, 1-n, -n], 1)
[round(A036242(n).n(100)) for n in (1..20)] # Peter Luschny, Sep 14 2014

Formula

Recurrence equation: a(n+1) = (2*n+2)*a(n) + a(n-1) with a(0) = 1 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n-1)/2)} 2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k,k+1). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) = 2^(n-1)*n!*hypergeometric([(1-n)/2, 1-n/2],[2, 1-n, -n], 1) for n>=2. - Peter Luschny, Sep 14 2014

A305459 a(0) = 1, a(1) = 3, a(n) = 3*n*a(n-1) + a(n-2).

Original entry on oeis.org

1, 3, 19, 174, 2107, 31779, 574129, 12088488, 290697841, 7860930195, 236118603691, 7799774851998, 281028013275619, 10967892292601139, 460932504302523457, 20752930585906156704, 996601600627798045249, 50847434562603606464403
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2018

Keywords

Comments

Let S(i,j,n) denote a sequence of the form a(0) = 1, a(1) = i, a(n) = i*n*a(n-1) + j*a(n-2). Then S(i,j,n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*i^(n-2*k)*j^k.

Links

Crossrefs

Cf. A001040, A036243, A213190, A305460.

Programs

  • GAP
    List([0..20],n->Sum([0..Int(n/2)],k->((Factorial(n-k))/(Factorial(k))*Binomial(n-k,k)*3^(n-2*k)))); # Muniru A Asiru, Jun 01 2018
  • Maple
    a:=proc(n) option remember: if n=0 then 1 elif n=1 then 3 elif n>=2 then 3*n*procname(n-1)-procname(n-2) fi; end:
seq(a(n),n=0..20); # Muniru A Asiru, Jun 01 2018
  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==3,a[n]==3n a[n-1]+a[n-2]},a,{n,20}] (* Harvey P. Dale, Aug 27 2019 *)
  • PARI
    {a(n) = sum(k=0, n/2, ((n-k)!/k!)*binomial(n-k,k)*3^(n-2*k))}

Formula

a(n) = Sum_{k=0..floor(n/2)} ((n-k)!/k!)*binomial(n-k,k)*3^(n-2*k).
a(n) ~ BesselI(0, 2/3) * n! * 3^n. - Vaclav Kotesovec, Jun 03 2018

A305401 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).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 3, 0, 1, 3, 9, 10, 1, 1, 4, 19, 56, 43, 0, 1, 5, 33, 174, 457, 225, 1, 1, 6, 51, 400, 2107, 4626, 1393, 0, 1, 7, 73, 770, 6433, 31779, 55969, 9976, 1, 1, 8, 99, 1320, 15451, 129060, 574129, 788192, 81201, 0, 1, 9, 129, 2086, 31753, 387045, 3103873, 12088488, 12667041, 740785, 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,  3,   9,   19,   33,    51, ...
   0, 10,  56,  174,  400,   770, ...
   1, 43, 457, 2107, 6433, 15451, ...

Links

Crossrefs

Columns k=0-3 give A059841, A001040(n+1), A036243, A305459.
Rows n=0-2 give A000012, A001477, A058331.
Main diagonal gives A305465.
Cf. A305466.

Formula

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

A246658 Triangle read by rows: T(n,k) = K(n,1)*I(k,1) - (-1)^(n+k)*I(n,1)* K(k,1), where I(n,x) and K(n,x) are Bessel functions; 0<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 9, 4, 1, 0, 56, 25, 6, 1, 0, 457, 204, 49, 8, 1, 0, 4626, 2065, 496, 81, 10, 1, 0, 55969, 24984, 6001, 980, 121, 12, 1, 0, 788192, 351841, 84510, 13801, 1704, 169, 14, 1, 0, 12667041, 5654440, 1358161, 221796, 27385, 2716, 225, 16, 1, 0
Offset: 0

Views

Author

Peter Luschny, Sep 14 2014

Keywords

Examples

			     0;
     1,      0;
     2,      1,     0;
     9,      4,     1,     0;
    56,     25,     6,     1,    0;
   457,    204,    49,     8,    1,   0;
  4626,   2065,   496,    81,   10,   1,  0;
55969,  24984,  6001,   980,  121,  12,  1, 0;
788192, 351841, 84510, 13801, 1704, 169, 14, 1, 0;

Crossrefs

Cf. A036242, A036243.

Programs

  • Maple
    T := (n, k) -> BesselK(n,1)*BesselI(k,1) - (-1)^(n+k)*BesselI(n,1)* BesselK(k,1);
seq(lprint(seq(round(evalf(T(n, k), 99)), k=0..n)), n=0..8);
  • Sage
    T = lambda n, k: bessel_K(n,1)*bessel_I(k,1) - (-1)^(n+k)*bessel_I(n,1)* bessel_K(k,1)
for n in range(9): [T(n,k).n().round() for k in (0..n)]

Formula

T(n, 0) = A036243(n-1) for n>=2.
T(n, 1) = A036242(n-1) for n>=2.

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).

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 1-6 of 6 results.

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

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