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.

A178078 Sequence with a (1,-1) Somos-4 Hankel transform.

Original entry on oeis.org

1, 0, 1, 1, 4, 12, 42, 147, 527, 1914, 7039, 26159, 98110, 370919, 1412211, 5410273, 20841886, 80685792, 313747624, 1224895416, 4799435482, 18867423751, 74394859297, 294152650731, 1166021396660, 4632969618849, 18448290723435
Offset: 0

Views

Author

Paul Barry, May 19 2010

Keywords

Comments

Hankel transform is A178079.

Links

Programs

  • Mathematica
    Table[Sum[(Binomial[n-k, k]/(n-2*k+1))*Sum[Binomial[k, j]*Binomial[n-k-j-1, n-2*k-j]*3^(n-2*k-j)*(-2)^j*1^(k-j), {j, 0, k}], {k, 0, Floor[n/2]}] + ((1 + (-1)^n)*(2/3)^(n/2))/2, {n, 0, 50}]  (* G. C. Greubel, Sep 18 2018 *)
  • PARI
    a(n) = sum(k=0,floor(n/2), sum(j=0,k, (binomial(n-k,k)/(n-2*k+1)) *binomial(k,j)*binomial(n-k-j-1,n-2*k-j)*3^(n-2*k-j)*(-2)^j));
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, Sep 18 2018

Formula

a(n) = Sum_{k=0..floor(n/2)} ( (C(n-k,k)/(n-2k+1))*Sum_{i=0..k} C(k,i)*C(n-k-i-1,n-2*k-i)*3^(n-2*k-i)*(-2)^i*1^(k-i) ).

A178578 Diagonal sums of second binomial transform of the Narayana triangle A001263.

Original entry on oeis.org

1, 3, 10, 34, 118, 417, 1497, 5448, 20063, 74649, 280252, 1060439, 4040413, 15488981, 59701236, 231236830, 899559100, 3513314664, 13770811198, 54152480421, 213585706927, 844723104691, 3349274471386, 13310603555085, 53012829376985, 211560158583657, 845856494229348, 3387782725245302, 13590698721293800, 54604853170818121, 219706932640295523
Offset: 0

Views

Author

Paul Barry, Dec 26 2010

Keywords

Comments

Hankel transform is the (1,-1) Somos-4 sequence A178079.

Links

Crossrefs

Cf. A025254.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 -3*x-x^2 - Sqrt(x^4+2*x^3+7*x^2-6*x+1))/(2*x^3))); // G. C. Greubel, Aug 14 2018
  • Mathematica
    Table[Sum[Sum[Binomial[n-k,j]*Binomial[j,k]*Binomial[j+1,k]*2^(n-k-j)/(k+1),{j,0,n-k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Mar 02 2014 *)
CoefficientList[Series[(1-3*x-x^2 -Sqrt[x^4+2*x^3+7*x^2-6*x+1])/(2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
  • PARI
    a(n)=sum(k=0,floor(n/2), sum(j=0,n-k,binomial(n-k,j)*binomial(j,k)*binomial(j+1,k)*2^(n-k-j)/(k+1)));
vector(22,n,a(n-1))

Formula

a(n) = A025254(n+2).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(n-k,j)*C(j,k)*C(j+1,k)*2^(n-k-j)/(k+1).
From Vaclav Kotesovec, Mar 02 2014: (Start)
Recurrence: (n+3)*a(n) = 3*(2*n+3)*a(n-1) - 7*n*a(n-2) - (2*n-3)*a(n-3) - (n-3)*a(n-4).
G.f.: (1 - 3*x - x^2 - sqrt(x^4 + 2*x^3 + 7*x^2 - 6*x + 1))/(2*x^3).
a(n) ~ (130-216*r-64*r^2-29*r^3) * sqrt(2*r^3+14*r^2-18*r+4) / (4 * sqrt(Pi) * n^(3/2) * r^n), where r = 1/6*(-3 + sqrt(3*(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))) - sqrt(-66 - 3*(1009 - 24*sqrt(183))^(1/3) - 3*(1009 + 24*sqrt(183))^(1/3) + 216*sqrt(3/(-11 + (1009 - 24*sqrt(183))^(1/3) + (1009 + 24*sqrt(183))^(1/3))))) = 0.23742047190096998... is the root of the equation r^4 + 2*r^3 + 7*r^2 - 6*r + 1 = 0.
(End)

A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.

Original entry on oeis.org

1, 1, -1, -4, -3, 19, 67, -40, -1243, -4299, 25627, 334324, 627929, -29742841, -372632409, 1946165680, 128948361769, 1488182579081, -52394610324649, -2333568937567764, -5642424912729707, 3857844273728205019
Offset: 1

Views

Author

Paul Barry, May 31 2010

Keywords

Comments

a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f.
1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-(201/361)x^2/(1-... where
1,4,-3/16,76/9,201/361,... are the x-coordinates of the multiples of z=(0,0)
on E:y^2-xy-y=x^3+x^2+x.

Links

Crossrefs

Cf. A006720.
(p, q) Somos-4 sequences: A171422, A174168, A174170, A174404, A174809, A174811, A174882, A178075, A178077, A178081, A178079, A178376, A178377, A178384, A178417, A178418, A178621, A178622, A178624, A178625, A178627, A178628, A178644, A184019, A184121, A188313, A188315, A352625.

Programs

  • Magma
    I:=[1,1,-1,-4]; [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[1] == 1, a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,30}] (* G. C. Greubel, Sep 18 2018 *)
  • PARI
    a(n)=local(E,z);E=ellinit([ -1,1,-1,1,0]);z=ellpointtoz(E,[0,0]); round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))
  • PARI
    m=30; v=concat([1,1,-1,-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, Sep 18 2018
  • PARI
    {a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* Michael Somos, Jul 05 2024 */
  • SageMath
    @CachedFunction
def a(n): # a = A178628
    if n<5: return (0,1,1,-1,-4)[n]
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 05 2024

Formula

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.
a(n) = -a(-n). a(n) = (-a(n-1)*a(n-4) +4*a(n-2)*a(n-3))/a(n-5) for all n in Z except n=5. - Michael Somos, Jul 05 2024

Extensions

Offset changed to 0. - Michael Somos, Jul 05 2024
Showing 1-3 of 3 results.

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