A178076
Sequence with a (4,-8) Somos-4 Hankel transform.
Original entry on oeis.org
1, 0, 2, -2, 8, -20, 64, -200, 656, -2184, 7408, -25456, 88480, -310464, 1098304, -3912896, 14026752, -50557184, 183110400, -666079872, 2432399104, -8914099840, 32772922112, -120844493056, 446793362944, -1656004554752
Offset: 0
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Table[If[n == 0, 1, Sum[(Binomial[n-k,k]/(n-2*k+1))* Sum[Binomial[k, j]*Binomial[n-k-j-1, n-2*k-j]*(-2)^(n-2*k-j)*2^k, {j, 0, k}], {k, 0, Floor[n/2]}] + (1 + (-1)^n)/2], {n, 0, 100}] (* G. C. Greubel, Sep 18 2018 *)
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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)*(-2)^(n-2*k-j)*2^k));
for(n=0,30, print1(a(n), ", ")) \\ G. C. Greubel, Sep 18 2018
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
(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.
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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
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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 *)
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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))
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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
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{a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* Michael Somos, Jul 05 2024 */
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@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
Showing 1-2 of 2 results.
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