A184019 -id:A184019 - cp's OEIS frontend

A184018 Expansion of c(x/(1-x-x^2)) / (1-x-x^2), c(x) the g.f. of A000108.

1, 2, 6, 19, 67, 254, 1017, 4236, 18168, 79680, 355635, 1609912, 7373401, 34102976, 159055728, 747211753, 3532452169, 16792693562, 80224098381, 384948157635, 1854469572120, 8965866981294, 43488834409737, 211569299607282

Offset: 0

Paul Barry, Jan 08 2011

Hankel transform is (9,-5) Somos-4 sequence A184019.

The radius of convergence r of the g.f. A(x) satisfies: r = (1-r-r^2)/4 = limit a(n)/a(n+1) = (sqrt(29)-5)/2 = 0.19258240... with A(r) = 1/(2*r) = (sqrt(29)+5)/4 = 2.59629120... - Paul D. Hanna, Sep 06 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A037027 := proc(n,m) add( binomial(m+k,m)*binomial(k,n-k-m),k=0..n-m) ; end proc:
A184018 := proc(n) add( A037027(n,k)*A000108(k),k=0..n) ; end proc:
seq(A184018(n),n=0..10) ; # R. J. Mathar, Jan 11 2011

CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 6 x^3 + x^4])/(2 x (1 - x - x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

{a(n)=polcoeff((1-sqrt(1-4*x/(1-x-x^2 +O(x^(n+2)))))/(2*x),n)} /* Paul D. Hanna, Sep 06 2011 */

G.f.: ( 1 - x - x^2 - sqrt((1 - x - x^2)*(1 - 5*x - x^2)) )/( 2*x*(1 - x - x^2) ).
G.f.: 1/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1-... (continued fraction).
a(n) = Sum_{k=0..n} (Sum_{j=0..n-k} binomial(k+j,k)*binomial(j,n-k-j))*A000108(k) = Sum_{k=0..n} A037027(n,k)*A000108(k).
G.f. satisfies A(x) = 1/(1-x-x^2) + x*A(x)^2. - Paul D. Hanna, Sep 06 2011
Conjecture: (n+1)*a(n) + 2*(1-3*n)*a(n-1) + 3*(n-1)*a(n-2) + 2*(3*n-5)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011
a(n) ~ (27+5*sqrt(29)) * sqrt(54*sqrt(29)-290) * (5+sqrt(29))^n / (sqrt(Pi) * n^(3/2) * 2^(n+5)). - Vaclav Kotesovec, Feb 04 2014

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

Paul Barry, May 31 2010

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.

G. C. Greubel, Table of n, a(n) for n = 1..160
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

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.

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

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 *)

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))

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

{a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* Michael Somos, Jul 05 2024 */

@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

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

Offset changed to 0. - Michael Somos, Jul 05 2024

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A184018 Expansion of c(x/(1-x-x^2)) / (1-x-x^2), c(x) the g.f. of A000108.

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

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