A173992 -id:A173992 - cp's OEIS frontend

A173993 Sequence whose Hankel transform is the Somos (4) sequence.

1, 2, 6, 17, 50, 146, 430, 1267, 3746, 11091, 32900, 97716, 290586, 864980, 2577032, 7683397, 22922874, 68427057, 204362172, 610604629, 1825092080, 5457016431, 16321318264, 48828168580, 146112907266, 437319580738, 1309158060068

Offset: 0

Paul Barry, Mar 04 2010

Hankel transform is A006720(n+3).

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

Cf. A006720, A173992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((Sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3))-2*x^2+4*x-1)/(2*x*(1-4*x+3*x^2)))); // G. C. Greubel, Sep 22 2018

CoefficientList[Series[(Sqrt[(1-2*x)*(1-2*x-4*x^2+4*x^3)]-2*x^2+4*x-1)/( 2 x*(1 - 4 x + 3 x^2)), {x, 0, 50}], x] (* G. C. Greubel, Sep 22 2018 *)

my(x='x+O('x^50)); Vec((sqrt((1-2*x)*(1-2*x-4*x^2+4*x^3))-2*x^2+4*x-1)/(2*x*(1-4*x+3*x^2))) \\ G. C. Greubel, Sep 22 2018

G.f.: (sqrt((1-2x)*(1-2x-4x^2+4x^3))-2x^2+4x-1)/(2x*(1-4x+3x^2)).

Conjecture: (n+1)*a(n) +2*(-4*n-1)*a(n-1) +(19*n-5)*a(n-2) -36*a(n-3) +8*(-7*n+26)*a(n-4) +2*(34*n-143)*a(n-5) +24*(-n+5)*a(n-6)=0. - R. J. Mathar, Oct 10 2014

A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

Original entry on oeis.org

1, 2, 3, 14, 69, 413, 7222, 90211, 2577626, 127385577, 5092018073, 655664812074, 78618294607139, 13276948495989478, 5995083279193033837, 1895278734817024984181, 1542333923096758721461086, 1867485777936169465836858947, 2020248742951852823878208914098, 7078136335206254534330825538868049

Offset: 0

Thomas Scheuerle, Oct 22 2024

nonn

Consider the sequence s(k) with ordinary generating function: 1/(1-d(0)*x/(1-d(1)*x/(1-d(2)*x/(...)))), the Hankel sequence transform of s(k) is A006720 starting with the third term.
This is a special case of a more general theorem: Consider the sequence h(k) with generating function 1/(1-c(0)*x/(1-c(1)*x/(1-c(2)*x/(...)))), if c(2*k+1) = (c(2*k-2)*c(2*k-1) + t)/(c(2*k-4)*c(2*k-3)*c(2*k-2)^2*c(2*k-1)^2*c(2*k)) for all k with c(< 0) = 1, then the Hankel sequence transform of h(k) satisfies a Somos-4 A(1, t) recurrence.
If we would change the start condition into d(0..4) = {1,1,-1,-2,(5/2)}, the expansion of the continued fraction generating function would give us A171416, its Hankel sequence transform is again A006720. There exist infinitely many sequences with the same Hankel sequence transform.

Cf. A006720, A028940.

The Hankel transform is directly related to A006720: A157002, A157003, A160702, A171416, A173992, A173993, A254314.

d(n) = if(n<5, [1,1,1,2,1][n+1], (d(n-3)*d(n-2)+1)/(d(n-5)*d(n-4)*d(n-3)^2*d(n-2)^2*d(n-1)))
a(n) = numerator(d(2*n+1))

a(n) = -numerator(ellmul(ellinit([0, 3, -1, 2, 0]), [-1,0], 2*n+1)[1])

a(n) = A006720(n+1)*A006720(n+3).
denominator(d(2*n+1)) = A006720(n+2)^2.
-a(n)/A006720(n+3)^2 are the x-coordinates of (2*n+1) times [-1,0] on the curve y^2 - y = x^3 + 3*x^2 + 2*x. "Times" means here the multiplication under the elliptic group law.

A348593 Triangle read by rows: T(n,m) = Sum_{j=0..min(m,n-m)} C(2j,j)C(n-2j-1,m-j)C(n-m,j)/(j+1).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 1, 6, 7, 1, 1, 8, 18, 6, 1, 1, 10, 34, 30, 7, 1, 1, 12, 55, 88, 33, 8, 1, 1, 14, 81, 195, 145, 42, 9, 1, 1, 16, 112, 366, 460, 184, 52, 10, 1, 1, 18, 148, 616, 1146, 763, 248, 63, 11, 1, 1, 20, 189, 960, 2422, 2544, 1060, 324, 75, 12, 1, 1, 22, 235, 1413, 4558, 6916, 4282, 1490, 413, 88, 13, 1

Offset: 0

Vladimir Kruchinin, Jan 25 2022

			Triangle begins
  1;
  1;
  1,  2;
  1,  4,  1;
  1,  6,  7,  1;
  1,  8, 18,  6,  1;
  1, 10, 34, 30,  7,  1;
  1, 12, 55, 88, 33,  8,  1;

Row sums give A173992.

Cf. A000108, A307374.

T(n,m):=sum(binomial(2*j,j)*binomial(n-2*j-1,m-j)*binomial(n-m,j)/(j+1), j,0,min(m,n-m));

G.f.: (1-sqrt(1-4*x^2*y*(1-x*y)/(1-x-x*y)))/(2*x^2*y).

Sum_{m>=0} (-1)^m * T(n,m) = A307374(n). - Alois P. Heinz, Jan 26 2022

cp's OEIS Frontend

A173993 Sequence whose Hankel transform is the Somos (4) sequence.

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A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

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PARI

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A348593 Triangle read by rows: T(n,m) = Sum_{j=0..min(m,n-m)} C(2j,j)C(n-2j-1,m-j)C(n-m,j)/(j+1).

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cp's OEIS Frontend

A173993 Sequence whose Hankel transform is the Somos (4) sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A377264 Consider the recurrence d(k) = (d(k-3)*d(k-2) + 1)/(d(k-5)*d(k-4)*d(k-3)^2*d(k-2)^2*d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2*n+1)).

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PARI

PARI

Formula

A348593 Triangle read by rows: T(n,m) = Sum_{j=0..min(m,n-m)} C(2j,j)*C(n-2j-1,m-j)*C(n-m,j)/(j+1).

Views

Author

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Examples

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Maxima

Formula

A377264 Consider the recurrence d(k) = (d(k-3)d(k-2) + 1)/(d(k-5)d(k-4)d(k-3)^2d(k-2)^2d(k-1)), with d(0..4) = {1,1,1,2,1}. a(n) = numerator(d(2n+1)).

A348593 Triangle read by rows: T(n,m) = Sum_{j=0..min(m,n-m)} C(2j,j)C(n-2j-1,m-j)C(n-m,j)/(j+1).