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.

A168050 Hankel transform of A168049.

Original entry on oeis.org

1, 1, 0, -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, -7, -7, -8, -8, -9, -9, -10, -10, -11, -11, -12, -12, -13, -13, -14, -14, -15, -15, -16, -16, -17, -17, -18, -18, -19, -19, -20, -20, -21, -21, -22, -22, -23, -23, -24, -24, -25, -25, -26, -26, -27, -27
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Links

Crossrefs

Cf. A168049.

Programs

  • Magma
    [(-1)^n/4-(2*n-3)/4+Binomial(1,n)-Binomial(0,n): n in [0..80]]; // Vincenzo Librandi, Jul 08 2016
  • Mathematica
    Join[{1,1,b=0},a=0;Table[c=b+2*a+n;a=b;b=c,{n,-1,60}]] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
CoefficientList[Series[(1 - 2 x^2 - x^3 + x^4)/((1 + x) (1 - x)^2), {x, 0, 100}], x] (* G. C. Greubel, Jul 07 2016 *)
Table[(-1)^n/4 - (2 n - 3)/4 + Binomial[1, n] - Binomial[0, n], {n, 0, 80}] (* Vincenzo Librandi, Jul 08 2016 *)
LinearRecurrence[{1,1,-1},{1,1,0,-1,-1},60] (* Harvey P. Dale, Dec 05 2018 *)
  • PARI
    Vec((1-2*x^2-x^3+x^4)/((1+x)*(1-x)^2) + O(x^99)) \\ Altug Alkan, Jul 08 2016

Formula

G.f.: (1 - 2x^2 - x^3 + x^4)/((1+x)(1-x)^2).
a(n) = + 1*a(n-1) + 1*a(n-2) - 1*a(n-3). - Joerg Arndt, Apr 02 2011
a(n) = (-1)^n/4 -(2n-3)/4 + C(1,n) - C(0,n).
E.g.f.: (4*x + exp(-x) - (2*x - 3)*exp(x))/4. - Ilya Gutkovskiy, Jul 08 2016

A168051 Expansion of (1+x+sqrt(1-2x-3x^2))/2.

Original entry on oeis.org

1, 0, -1, -1, -2, -4, -9, -21, -51, -127, -323, -835, -2188, -5798, -15511, -41835, -113634, -310572, -853467, -2356779, -6536382, -18199284, -50852019, -142547559, -400763223, -1129760415, -3192727797, -9043402501, -25669818476
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

A signed variant of the Motzkin numbers A001006. Hankel transform is A168052.

Examples

			G.f. = 1 - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 21*x^7 - 51*x^8 - 127*x^9 + ...

Links

Crossrefs

Cf. A168049, A166587, A167022.
Cf. A001006, A005043.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1 + x + Sqrt[1 - 2 x - 3 x^2]) / 2, {x, 0, n}] (* Michael Somos, Jan 25 2014 *)
  • PARI
    {a(n) = polcoeff( (1 + x + sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2, n)}; /* Michael Somos, Jan 25 2014 */

Formula

D-finite with recurrence: n*a(n) -(2n-3)*a(n-1) -3*(n-3)*a(n-2)=0 if n>2. - R. J. Mathar, Dec 20 2011 [Edited by Michael Somos, Jan 25 2014]
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 25 2014
G.f.: 1 + x - (x + x^2) / (1 + x - (x + x^2) / (1 + x - ...)). - Michael Somos, Mar 27 2014
Convolution inverse of A005043. - Michael Somos, Mar 27 2014
a(n) ~ -3^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 05 2018
From Gennady Eremin, Feb 25 2021: (Start)
For n > 1, a(n) = A167022(n) / 2.
G.f.: (1 + x + A(x)) / 2, where A(x) is the g.f. of A167022. (End)

A168055 Expansion of 2 - x - sqrt(1-2x-3x^2).

Original entry on oeis.org

1, 0, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

Hankel transform is A168054.

Examples

			G.f. = 1 + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ...

Links

Crossrefs

Cf. A168049.
Cf. A126068, A007971. [R. J. Mathar, Nov 18 2009]

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2 - x - Sqrt[1 - 2 x - 3 x^2], {x, 0, n}] (* Michael Somos, Jan 25 2014 *)
  • PARI
    {a(n) = polcoeff( 2 - x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n)), n)} /* Michael Somos, Jan 25 2014 */

Formula

a(n+2) = 2*A001006(n).
a(n) = 0^n + 2*Sum_{k=0..floor((n-2)/2)} C(n-2,2k)*A000108(k).
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 25 2014
D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +3*(-n+3)*a(n-2)=0. - R. J. Mathar, Nov 19 2014

Extensions

Name corrected by Michael Somos, Mar 23 2012

A168073 Expansion of 1 + 3*(1-x-sqrt(1-2*x-3*x^2))/2.

Original entry on oeis.org

1, 0, 3, 3, 6, 12, 27, 63, 153, 381, 969, 2505, 6564, 17394, 46533, 125505, 340902, 931716, 2560401, 7070337, 19609146, 54597852, 152556057, 427642677, 1202289669, 3389281245, 9578183391, 27130207503, 77009455428, 219023318406, 624069834627, 1781228354487
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Comments

Hankel transform is A168072. a(n+2)=3*A000106(n). Another variant is A168076.

Links

Crossrefs

Cf. A168055, A168049.
Cf. A000106, A168072, A168076.

Programs

  • Mathematica
    CoefficientList[Series[1 + 3*(1 - x - Sqrt[1 - 2*x - 3*x^2])/2, {x, 0, 50}], x] (* G. C. Greubel, Jul 09 2016 *)

Formula

a(n) = 0^n+3*Sum_{k=0..floor((n-2)/2)} C(n-2,2k)*A000108(k).
D-finite with recurrence: a(n) = ((2*n-3)*a(n-1)+(3*n-9)*a(n-2))/n for n>=3, a(0)=1, a(1)=0, a(2)=3. - Sergei N. Gladkovskii, Jul 16 2012

A168076 Expansion of 1 - 3*(1-x-sqrt(1-2*x-3*x^2))/2.

Original entry on oeis.org

1, 0, -3, -3, -6, -12, -27, -63, -153, -381, -969, -2505, -6564, -17394, -46533, -125505, -340902, -931716, -2560401, -7070337, -19609146, -54597852, -152556057, -427642677, -1202289669, -3389281245, -9578183391, -27130207503, -77009455428, -219023318406
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Comments

For n>0, a(n) = -3*A168049(n). Hankel transform is A168075. Another variant is A168073.

Links

Crossrefs

Cf. A168049, A168073, A168075.

Programs

  • Maple
    f:= gfun:-rectoproc({(3*n-3)*a(n)+(1+2*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 0, a(2) = -3},a(n),remember):
map(f, [$0..60]); # Robert Israel, May 13 2018
  • Mathematica
    CoefficientList[Series[1 - 3*(1 - x - Sqrt[1 - 2*x - 3*x^2])/2, {x,0,50}] , x] (* G. C. Greubel, Jul 09 2016 *)
  • PARI
    x='x+O('x^99); Vec(1-3*(1-x-(1-2*x-3*x^2)^(1/2))/2) \\ Altug Alkan, May 13 2018
  • PARI
    A168076(n)=!n-3*sum(k=0,n\2-1, binomial(n-2,2*k)*binomial(2*k,k)/(k+1)) \\ M. F. Hasler, May 13 2018

Formula

a(n) = 0^n - 3*Sum_{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.
D-finite with recurrence: n*a(n) + (-2*n+3)*a(n-1) + 3*(-n+3)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
Recurrence (for n >= 3) follows from the differential equation (3*x^2+2*x-1)*y' - (3*x+1)*y = 3*x-1 satisfied by the g.f. - Robert Israel, May 13 2018
a(n) ~ -3^(n+1/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
a(n) = -A168073(n) <= 0 for n >= 1. - M. F. Hasler, May 13 2018

Extensions

Comment corrected by Vaclav Kotesovec, Dec 03 2014
Showing 1-5 of 5 results.

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