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 1-4 of 4 results.

A060196 Decimal expansion of 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + ...

Original entry on oeis.org

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

Views

Author

Evan Michael Adams (evan(AT)tampabay.rr.com), Simon Plouffe, Mar 21 2001

Keywords

Examples

			1.410686134642447997690824711419115041323478...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.2, p. 423.

Links

Crossrefs

Cf. A001147, A112293, A286286.

Programs

  • Mathematica
    RealDigits[ Sqrt[E*Pi/2] * Erf[1/Sqrt[2]], 10, 107] // First
(* or *) 1/Fold[Function[2*#2-1+(-1)^#2*#2/#1], 1, Reverse[Range[100]]] // N[#, 107]& // RealDigits // First (* Jean-François Alcover, Mar 07 2013, updated Sep 19 2014 *)
  • PARI
    { default(realprecision, 20080); x=2^(-1/2)*exp(1/2)*sqrt(Pi)*(1 - erfc(1/sqrt(2))); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060196.txt", n, " ", d)); } \\ Harry J. Smith, Jul 02 2009

Formula

c = sqrt(e*Pi/2)*erf(1/sqrt(2)), or 2^(-1/2)*exp(1/2)*sqrt(Pi)*(1 - erfc(1/sqrt(2))). - Michael Kleber, Mar 21 2001
From Peter Bala, Feb 09 2024: (Start)
Generalized continued fraction expansion:
c = 1/(1 - 1/(4 - 3/(6 - 5/(8 - 7/(10 - 9/(12 - ... )))))). See A286286.
c/(1 + c) = Sum_{n >= 0} (2*n-1)!!/(A112293(n) * A112293(n+1)) = 1/(1*2) + 1/(2*7) + 3/(7*36) + 15/(36*253) + 105/(253*2278) + ... = 0.5851803411..., a rapidly converging series. (End)
Equals Sum_{n >= 0} ((n - 1)*(n + 1)!*2^(n + 1))/(2*n)!. - Antonio Graciá Llorente, Feb 13 2024

Extensions

More terms from Vladeta Jovovic, Mar 27 2001

A112292 An invertible triangle of ratios of double factorials.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 15, 5, 1, 105, 105, 35, 7, 1, 945, 945, 315, 63, 9, 1, 10395, 10395, 3465, 693, 99, 11, 1, 135135, 135135, 45045, 9009, 1287, 143, 13, 1, 2027025, 2027025, 675675, 135135, 19305, 2145, 195, 15, 1, 34459425, 34459425, 11486475, 2297295, 328185, 36465, 3315, 255, 17, 1
Offset: 0

Views

Author

Paul Barry, Sep 01 2005

Keywords

Comments

As a square array read by antidiagonals, column k has e.g.f. (1/(1-2x)^(1/2))*(1/(1-2x))^k. - Paul Barry, Sep 04 2005
Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) is this sequence and T(n, k, 3) = A136214. - Peter Luschny, Jun 01 2009, revised Jun 18 2019

Examples

			Triangle begins
      1;
      1,     1;
      3,     3,    1;
     15,    15,    5,  1;
    105,   105,   35,  7,  1;
    945,   945,  315, 63,  9,  1;
  10395, 10395, 3465,693, 99, 11, 1;
Inverse is A112295, which begins
   1;
  -1,  1;
   0, -3,  1;
   0,  0, -5,  1;
   0,  0,  0, -7,  1;
   0,  0,  0,  0, -9,  1;
Similar results arise for higher factorials.

Crossrefs

Columns include A001147, A051577, A051579.
Row sums are A112293.
Diagonal sums are A112294.
Cf. A094587 (p=1), this sequence (p=2), A136214 (p=3).

Programs

  • Mathematica
    T[n_, k_] := If[k <= n, (2n-1)!!/(2k-1)!!, 0];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

Formula

T(n, k)=if(k<=n, (2n-1)!!/(2k-1)!!, 0);
T(n, k)=if(k<=n, n!*C(2n, n)2^(k-n)/(k!*C(2k, k)), 0);
T(n, k)=if(k<=n, 2^(n-k)(n-1/2)!/(k-1/2)!, 0);
T(n, k)=if(k<=n, (n+1)!*C(n)2^(k-n)/((k+1)!*C(k)), 0).

A178120 Coefficient array of orthogonal polynomials P(n,x)=(x-2n)*P(n-1,x)-(2n-3)*P(n-2,x), P(0,x)=1,P(1,x)=x-2.

Original entry on oeis.org

1, -2, 1, 7, -6, 1, -36, 40, -12, 1, 253, -326, 131, -20, 1, -2278, 3233, -1552, 324, -30, 1, 25059, -38140, 20678, -5260, 675, -42, 1, -325768, 523456, -310560, 90754, -14380, 1252, -56, 1, 4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1
Offset: 0

Views

Author

Paul Barry, May 20 2010

Keywords

Comments

Inverse is A178121. First column is A112293 signed.

Examples

			Triangle begins
1,
-2, 1,
7, -6, 1,
-36, 40, -12, 1,
253, -326, 131, -20, 1,
-2278, 3233, -1552, 324, -30, 1,
25059, -38140, 20678, -5260, 675, -42, 1,
-325768, 523456, -310560, 90754, -14380, 1252, -56, 1,
4886521, -8205244, 5223602, -1694244, 312059, -33866, 2135, -72, 1
Production matrix of inverse is
2, 1,
1, 4, 1,
0, 3, 6, 1,
0, 0, 5, 8, 1,
0, 0, 0, 7, 10, 1,
0, 0, 0, 0, 9, 12, 1,
0, 0, 0, 0, 0, 11, 14, 1,
0, 0, 0, 0, 0, 0, 13, 16, 1,
0, 0, 0, 0, 0, 0, 0, 15, 18, 1

Programs

  • Maple
    A178120 := proc(n,k)
    if n = k then
        1;
    elif n = 1 and k = 0 then
        -2 ;
    elif k < 0 or k > n then
        0 ;
    else
        -2*n*procname(n-1,k)+procname(n-1,k-1)-(2*n-3)*procname(n-2,k) ;
    end if;
end proc: # R. J. Mathar, Dec 03 2014
  • Mathematica
    P[0, _] = 1;
P[1, x_] := x - 2;
P[n_, x_] := P[n, x] = (x-2n) P[n-1, x] - (2n-3) P[n-2, x];
T[n_] := Module[{x}, CoefficientList[P[n, x], x]];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Aug 06 2023 *)

A368790 a(n) = (3*n-2)!!! * Sum_{k=0..n} 1/(3*k-2)!!!.

Original entry on oeis.org

1, 2, 9, 64, 641, 8334, 133345, 2533556, 55738233, 1393455826, 39016763129, 1209519657000, 41123668338001, 1521575728506038, 60863029140241521, 2617110253030385404, 120387071639397728585, 5898966510330488700666, 306746258537185412434633
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2024

Keywords

Crossrefs

Row sums of A136214.
Cf. A007559, A112293, A368791.

Programs

  • PARI
    a007559(n) = prod(k=1, n, 3*k-2);
a(n) = a007559(n)*sum(k=0, n, 1/a007559(k));

Formula

a(n) = (3*n-2) * a(n-1) + 1.
Showing 1-4 of 4 results.

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