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.

A249057 Triangular array: Row n shows the coefficients of polynomials p(n,x) defined in Comments.

Original entry on oeis.org

1, 4, 1, 5, 4, 1, 24, 11, 4, 1, 35, 52, 18, 4, 1, 192, 123, 84, 26, 4, 1, 315, 660, 285, 120, 35, 4, 1, 1920, 1545, 1500, 545, 160, 45, 4, 1, 3465, 9180, 4680, 2820, 930, 204, 56, 4, 1, 23040, 22005, 27180, 11220, 4740, 1470, 252, 68, 4, 1, 45045, 142380
Offset: 0

Views

Author

Clark Kimberling, Oct 20 2014

Keywords

Comments

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + (n + 1)/f(n-1,x), where f(0,x) = 1.
Row sums give A249059(n) for n >= 1.
First column is A249060 (n-th term = n!! for n >= 0).

Examples

			f(0,x) = 1/1, so that p(0,x) = 1
f(1,x) = (4 + x)/1, so that p(1,x) = 4 + x;
f(2,x) = (5 + 4 x + x^2)/(1 + x), so that p(2,x) = 5 + 4 x + x^2.
First 6 rows of the triangle of coefficients:
1
4    1
5    4     1
24   11    4    1
35   52    18   4    1
192  123   84   26   4   1

Links

Crossrefs

Cf. A249059, A249060.

Programs

  • Mathematica
    z = 12; f[x_, n_] := x + (n+3)/f[x, n - 1];
f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}];
u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]];
Flatten[CoefficientList[u, x]] (* A249057 sequence *)
  • PARI
    f(n) = if (n, x + (n + 3)/f(n-1), 1);
row(n) = Vecrev(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022

A249060 Column 1 of the triangular array at A249057.

Original entry on oeis.org

1, 4, 5, 24, 35, 192, 315, 1920, 3465, 23040, 45045, 322560, 675675, 5160960, 11486475, 92897280, 218243025, 1857945600, 4583103525, 40874803200, 105411381075, 980995276800, 2635284526875, 25505877196800, 71152682225625, 714164561510400, 2063427784543125
Offset: 0

Views

Author

Clark Kimberling, Oct 20 2014

Keywords

Examples

			First 3 rows from A249057:
1
4    1
5    4    1,
so that a(0) = 1, a(1) = 4, a(2) = 5.

Links

Crossrefs

Cf. A178647, A249057, A309491.

Programs

  • Mathematica
    z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}];
u = Numerator[t]; v1 = Flatten[CoefficientList[u, x]]; (* A249057 *)
v2 = u /. x -> 1  (* A249059 *)
v3 = u /. x -> 0  (* A249060 *)
  • PARI
    f(n) = if (n, x + (n + 3)/f(n-1), 1);
a(n) = polcoef(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022

Formula

From Derek Orr, Oct 21 2014: (Start)
a(2*n) = (2*n+3)*(2*n+1)!!/3, for n > 0.
a(2*n+1) = (n+2)!*2^(n+1), for n > 0.
For n > 2, if n is even, a(n)/[(n+1)*(n-1)*(n-3)*...*7*5] = n + 3 and if n is odd, a(n)/[(n+1)*(n-1)*(n-3)*...*6*4] = n + 3. (End)
a(n) = gcd_2((n+3)!,(n+3)!!), where gcd_2(b,c) denotes the second-largest common divisor of non-coprime integers b and c, as defined in A309491. - Lechoslaw Ratajczak, Apr 15 2021
D-finite with recurrence: a(n) - (3+n)*a(n-2) = 0. - Georg Fischer, Nov 25 2022
Sum_{n>=0} 1/a(n) = 3*sqrt(e*Pi/2)*erf(1/sqrt(2)) + 2*sqrt(e) - 6, where erf is the error function. - Amiram Eldar, Dec 10 2022

A291856 a(0) = -1, a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n > 1.

Original entry on oeis.org

-1, 1, 0, 2, 2, 10, 20, 80, 220, 860, 2840, 11440, 42680, 179960, 734800, 3254240, 14276240, 66344080, 309040160, 1503233600, 7374996640, 37439668640, 192314598080, 1015987308160, 5439223064000, 29822918459840, 165803495059840, 941199375015680
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2017

Keywords

Links

Crossrefs

Cf. A000085, A000932, A249059.

Programs

  • GAP
    a:=[-1,1];; for n in [3..10^2] do a[n]:=a[n-1]+(n-2)*a[n-2]; od; a;  # Muniru A Asiru, Sep 07 2017
  • Mathematica
    RecurrenceTable[{a[0] == -1, a[1] == 1, a[n] == a[n-1] + (n-1)*a[n-2]}, a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 04 2017 *)
CoefficientList[Series[E^(x*(2 + x)/2) * (E^(1/2)*Sqrt[2*Pi]*(Erf[(1 + x)/Sqrt[2]] - Erf[1/Sqrt[2]]) - 1), {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Sep 05 2017 *)

Formula

a(n+4) = 2*A249059(n) for n >= 0.
E.g.f.: exp(x*(2+x)/2) * (exp(1/2) * sqrt(2*Pi) * (erf((1+x)/sqrt(2)) - erf(1/sqrt(2))) - 1). - Vaclav Kotesovec, Sep 05 2017
a(n) ~ (sqrt(Pi) * exp(1/2) * (1 - erf(1/sqrt(2))) - sqrt(2)/2) * n^(n/2) * exp(sqrt(n) - n/2 - 1/4). - Vaclav Kotesovec, Sep 05 2017
Showing 1-3 of 3 results.

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