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.

A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

Original entry on oeis.org

1, 2, 6, 18, 58, 192, 650, 2232, 7746, 27096, 95376, 337404, 1198546, 4272308, 15273888, 54744268, 196646922, 707747988, 2551624304, 9213416524, 33313656888, 120604436624, 437112790668, 1585877246424, 5759085911154
Offset: 0

Views

Author

Paul Barry, Feb 20 2009

Keywords

Comments

Hankel transform is A157005. Image of A000984 under Riordan array (1,x(1-x^2)).
Diagonal of rational function 1/(1 - x - y + x^3*y^2). - Seiichi Manyama, Mar 23 2023

Examples

			G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 192*x^5 + 650*x^6 + 2232*x^7 + ...

Links

Crossrefs

Cf. A157003, A360219, A360266.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-4*x+4*x^3) )); // G. C. Greubel, Feb 26 2019
  • Mathematica
    CoefficientList[Series[1/Sqrt[1-4*x*(1-x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m)*(1-x)^m / (1-2*x+x*O(x^n))^(2*m+1)), n)} \\ Paul D. Hanna, Sep 21 2013
  • PARI
    my(x='x+O('x^30)); Vec(1/sqrt(1-4*x+4*x^3)) \\ G. C. Greubel, Feb 26 2019
  • Sage
    (1/sqrt(1-4*x+4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Formula

G.f.: 1/sqrt(1 - 4*x*(1 - x^2)).
a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2)) *A000984(k)/2.
G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) * (1-x)^n / (1-2*x)^(2*n+1). - Paul D. Hanna, Sep 21 2013
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ (1/r)^n / (sqrt(Pi*n) * sqrt(3-8*r)), where r = 0.2695944364054... is the root of the equation 4*r*(1-r^2)=1. - Vaclav Kotesovec, Feb 13 2014
0 = a(n)*(16*a(n+1) - 32*a(n+3) + 10*a(n+4)) + a(n+1)*(-2*a(n+3)) + a(n+2)*(16*a(n+3) - 6*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Sep 03 2016

A162546 A Somos-4 variant: a(n) = (36*a(n-1)*a(n-3) - 68*a(n-2)^2)/a(n-4).

Original entry on oeis.org

1, 1, -16, -644, -40592, -4821056, 17059328, 2492895195136, 10659285907800064, 86296767700623425536, 1081586547380924161458176, -36649408809924048998874742784, -18144416387824430577315746611724288
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform of A162543, A162548.

Links

Crossrefs

Cf. A157005, A157101, A162547.

Programs

  • GAP
    a:=[1,1,-16,-644];; for n in [5..20] do a[n]:=(36*a[n-1]*a[n-3] - 68*a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
  • Magma
    I:=[1,1,-16,-644]; [n le 4 select I[n] else (36*Self(n-1) *Self(n-3) - 68*Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
  • Mathematica
    RecurrenceTable[{a[n]==(36*a[n-1]*a[n-3] - 68*a[n-2]^2)/a[n-4], a[0]==1, a[1]==1, a[2]==-16, a[3]==-644}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)
  • PARI
    m=20; v=concat([1,1,-16,-644], vector(m-4)); for(n=5, m, v[n] = (36*v[n-1]*v[n-3] -68*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
  • Sage
    def a(n):
    if (n==0): return 1
    elif (n==1): return 1
    elif (n==2): return -16
    elif (n==3): return -644
    else: return (36*a(n-1)*a(n-3) - 68*a(n-2)^2)/a(n-4)
[a(n) for n in (1..20)] # G. C. Greubel, Feb 23 2019

A157101 A Somos-4 variant.

Original entry on oeis.org

1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199
Offset: 0

Views

Author

Paul Barry, Feb 22 2009

Keywords

Comments

Hankel transform of A157100.

Links

Crossrefs

Cf. A051138.
Cf. A157005, A162546, A162547.

Programs

  • GAP
    a:=[1,-1,-5,-4];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
  • Magma
    I:=[1,-1,-5,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
  • Mathematica
    RecurrenceTable[{a[n]==(a[n-1]*a[n-3]+a[n-2]^2)/a[n-4], a[0]==1, a[1]==-1, a[2]==-5, a[3]==-4}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)
  • PARI
    m=20; v=concat([1,-1,-5,-4], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
  • Sage
    def a(n):
    if (n==0): return 1
    elif (n==1): return -1
    elif (n==2): return -5
    elif (n==3): return -4
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019

Formula

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), with a(0)=1, a(1)=-1, a(2)=-5, a(3)=-4.
a(n) = A051138(n+1) for all n in Z. - Michael Somos, Jul 17 2016

A165904 Somos-4 recurrence with a(i)=2^i for 0<=i<=3.

Original entry on oeis.org

1, 2, 4, 8, 32, 96, 448, 2944, 15104, 160768, 1565696, 16812032, 341250048, 5081473024, 128940408832, 4153114198016, 110562817540096, 6199850375708672, 294499013825921024, 17095312584100282368, 1743063839902067064832
Offset: 0

Views

Author

Jaume Oliver Lafont, Sep 29 2009

Keywords

Links

Crossrefs

Cf. A006720, A157005.

Programs

  • Magma
    I:=[1,2,4,8]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 18 2018
  • Mathematica
    RecurrenceTable[{a[n] == (a[n-1]*a[n-3] +a[n-2]^2)/a[n-4], a[0] == 1, a[1] == 2, a[2] == 4, a[3] == 8}, a, {n, 0, 30}] (* G. C. Greubel, Sep 18 2018 *)
  • PARI
    a(n)=if(n<4,[1,2,4,8][n+1],(a(n-1)*a(n-3)+a(n-2)^2)/a(n-4))

Formula

a(n) = 2^n*A006720(n).
a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4). - G. C. Greubel, Sep 18 2018

Extensions

"frac" keyword removed by Jaume Oliver Lafont, Oct 13 2009

A165905 Somos-4 recurrence with a(0)=1, a(1)=2, a(2)=4, a(3)=16.

Original entry on oeis.org

1, 2, 4, 16, 48, 224, 1472, 7552, 80384, 782848, 8406016, 170625024, 2540736512, 64470204416, 2076557099008, 55281408770048, 3099925187854336, 147249506912960512, 8547656292050141184, 871531919951033532416
Offset: 0

Views

Author

Jaume Oliver Lafont, Sep 29 2009

Keywords

Examples

			G.f. = 1 + 2*x + 4*x^2 + 16*x^3 + 224*x^4 + 1472*x^5 + 7552*x^6 + ... - _Michael Somos_, Sep 19 2018

Links

Crossrefs

Cf. A006720, A157005, A165904.

Programs

  • Magma
    I:=[1,2,4,16]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 18 2018
  • Mathematica
    RecurrenceTable[{a[n] == (a[n-1]*a[n-3] +a[n-2]^2)/a[n-4], a[0] == 1, a[1] == 2, a[2] == 4, a[3] == 16}, a, {n, 0, 30}] (* G. C. Greubel, Sep 18 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,(b*d+c^2)/a}; NestList[nxt,{1,2,4,16},20][[All,1]] (* Harvey P. Dale, Jul 28 2021 *)
  • PARI
    a(n)=if(n<4,[1,2,4,16][n+1],(a(n-1)*a(n-3)+a(n-2)^2)/a(n-4))

Formula

a(n) = 2^n*A006720(n+1) for all n in Z.
a(n) = (a(n-1)*a(n-3) +a(n-2)^2)/a(n-4). - G. C. Greubel, Sep 18 2018

Extensions

"frac" keyword removed by Jaume Oliver Lafont, Oct 13 2009
Showing 1-5 of 5 results.

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