A168051 -id:A168051 - cp's OEIS frontend

A168049 Expansion of (3 -x -sqrt(1-2x-3x^2))/2.

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, 73007772802, 208023278209

Offset: 0

Paul Barry, Nov 17 2009

A variant of the Motzkin numbers A001006. Hankel transform is A168050.
Essentially the same as A086246. - R. J. Mathar, Dec 20 2011
Alternatively, this sequence corresponds to the number of positive walks with n steps {-1,0,1} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 51*x^8 + ... - _Michael Somos_, Sep 26 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.

Cf. A001006, A086246, A168050, A168051.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3 -x - Sqrt(1-2*x-3*x^2))/2)); // G. C. Greubel, Sep 25 2018

CoefficientList[Series[(3-x-Sqrt[1-2*x-3*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Vec((3-x-sqrt(1-2*x-3*x^2))/2) \\ Charles R Greathouse IV, Dec 01 2016

D-finite with recurrence: n*a(n) +(3-2n)*a(n-1) +3(3-n)*a(n-2)=0. - R. J. Mathar, Dec 20 2011
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 31 2014
a(n) ~ 3^(n+1/2) / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
G.f.: 1 + x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Sep 23 2017

A167022 Expansion of sqrt(1 - 2x - 3x^2) in powers of x.

Original entry on oeis.org

1, -1, -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

Michael Somos, Oct 27 2009

sign

Sequence is to Motzkin numbers as A002420 is to Catalan numbers.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001006, A002426, A007971, A126068.

a[ n_] := SeriesCoefficient[ Sqrt[1 - 2 x - 3 x^2], {x, 0, n}] (* Michael Somos, Jan 25 2014 *)

{a(n) = polcoeff( sqrt(1 - 2*x - 3*x^2 + x * O(x^n)), n)}

D-finite with recurrence: n*a(n) = (2*n - 3)*a(n-1) + (3*n - 9)*a(n-2) for n>1.
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)) for all n in Z. - Michael Somos, Mar 23 2012
G.f.: sqrt(1 - 2*x - 3*x^2).
Convolution inverse of A002426. A007971(n) = -a(n) unless n=0. A126068(n) = -a(n) unless n=0 or n=1. A001006(n) = -a(n+2)/2 unless n=0 or n=1.
G.f.: A(x)=sqrt(1-2*a*x+((a)^2-4*b)*(x^2)) =1-a*x-2*b*x^2/G(0) ; G(k) = 1 - a*x - b*x^2/G(k+1). - Sergei N. Gladkovskii, Dec 05 2011
a=1;b=1;A(x)=(1-2*x-3*x^2)^(1/2)=1-x-2*x^2/G(0) ; G(k) = 1 - x - x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
G.f.: sqrt(1-2*x-3*(x^2))=1 - x/G(0) = (3*x+2)*G(0) - 1 ; G(k) = 1 - 2*x/(1 + x/(1 + x/(1 - 2*x/(1 - x/(2 - x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
a(n) ~ -3^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 05 2018
a(n) = 2*A168051(n), n>1. - R. J. Mathar, Jan 23 2020

A292440 Expansion of (1 - x + sqrt(1 - 2x - 3x^2))/2 in powers of x.

Original entry on oeis.org

1, -1, -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, -73007772802

Offset: 0

Seiichi Manyama, Sep 16 2017

sign

Apart from a(1) the same as A168051. - R. J. Mathar, Sep 18 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001006, A086246.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x +Sqrt(1-2*x-3*x^2))/2)); // G. C. Greubel, Aug 13 2018

CoefficientList[Series[(1-x +Sqrt[1-2*x-3*x^2])/2, {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)

x='x+O('x^50); Vec((1 - x + sqrt(1 - 2*x - 3*x^2))/2) \\ G. C. Greubel, Aug 13 2018

Convolution inverse of A001006.
Let f(x) = (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2).
G.f.: 1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(... (continued fraction).
G.f.: 1/f(x) = 1 - x - x^2*f(x).
a(n) = -A001006(n-2) for n > 1.
a(n) ~ -3^(n - 1/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 14 2018
D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +3*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 23 2020

A168052 Hankel transform of a Motzkin variant.

Original entry on oeis.org

1, -1, 2, -3, 3, -4, 5, -5, 6, -7, 7, -8, 9, -9, 10, -11, 11, -12, 13, -13, 14, -15, 15, -16, 17, -17, 18, -19, 19, -20, 21, -21, 22, -23, 23, -24, 25, -25, 26, -27, 27, -28, 29, -29, 30, -31, 31, -32, 33, -33, 34, -35, 35, -36, 37, -37, 38, -39, 39, -40, 41, -41, 42, -43

Offset: 0

Paul Barry, Nov 17 2009

Hankel transform of A168051.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,-1,-1).

Cf. A010892, A168051.

I:=[1,-1,2,-3]; [n le 4 select I[n] else - Self(n-1)-Self(n-3)- Self(n-4): n in [1..65]]; // Vincenzo Librandi, Jul 08 2016

LinearRecurrence[{-1, 0, -1, -1}, {1, -1, 2, -3}, 100] (* G. C. Greubel, Jul 07 2016 *)
CoefficientList[Series[(1 + x^2) / ((1 + x)^2 (1 - x + x^2)), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 08 2016 *)

G.f.: (1+x^2)/((1+x)^2*(1-x+x^2)).
a(n) = cos(Pi*n/3)/3 + sqrt(3)*sin(Pi*n/3)/9 + 2*(n+1)*(-1)^n/3.
a(n) = A010892(n)/3 + 2*(-1)^n*(n+1)/3. - R. J. Mathar, Sep 30 2012
E.g.f.: exp(-x)*(6 - 6*x + exp(3*x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 03 2023

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A168049 Expansion of (3 -x -sqrt(1-2*x-3*x^2))/2.

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A167022 Expansion of sqrt(1 - 2*x - 3*x^2) in powers of x.

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A292440 Expansion of (1 - x + sqrt(1 - 2*x - 3*x^2))/2 in powers of x.

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A168052 Hankel transform of a Motzkin variant.

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A168049 Expansion of (3 -x -sqrt(1-2x-3x^2))/2.

A167022 Expansion of sqrt(1 - 2x - 3x^2) in powers of x.

A292440 Expansion of (1 - x + sqrt(1 - 2x - 3x^2))/2 in powers of x.