A328380 -id:A328380 - cp's OEIS frontend

A242107 Reduced division polynomials associated with elliptic curve y^2 + x*y = x^3 - x^2 - x + 1 and multiples of point (0, 1).

0, 1, 1, 1, 1, -1, 2, -3, 1, -5, -7, -13, -16, 11, -57, 131, -113, 389, 670, 2311, 3983, 9, 23647, -81511, 140576, -484247, -833503, -5751815, -14871471, -17124617, -147165662, 710017141, -2273917871, 9644648819, 11396432249, 204006839259, 808162720720

Offset: 0

Michael Somos, Aug 15 2014

sign

This sequence is similar to Somos-5 (A006721).

For the elliptic curve "58a1" and point (0, 1) the multiple n*(0, 1) = ((3-(-1)^n)/2 * a(n+1)*a(n-1) / a(n)^2, a(n+2)^2 * a(n-4) / a(n)^3). - Michael Somos, Feb 23 2020

			a(9) = -5 and the point multiple 9*(0, 1) = (-14/(-5)^2, -169/(-5)^3).

Chai Wah Wu, Table of n, a(n) for n = 0..320
LMFDB, Elliptic Curve with LMFDB label 58.a1 (Cremona label 58a1).

Cf. A006721, A178622, A242108, A328380.

I:=[1,1,1,1,-1]; [0] cat [n le 5 select I[n] else (-Self(n-1)* Self(n-4) + Self(n-2)*Self(n-3))/Self(n-5): n in [1..30]]; // G. C. Greubel, Aug 05 2018

Join[{0}, RecurrenceTable[{a[n] == (-a[n-1]*a[n-4] + a[n-2]*a[n-3])/a[n-5], a[0] == 0, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == -1}, a, {n, 0, 50}]] (* G. C. Greubel, Aug 05 2018 *)

{a(n) = my(s=1, v); if( n<0, s=-1; n=-n); s^(n+1) * if( n, v = vector(n, k, 1); if( n<6, (-1)^(n>4), v[5] = -1; for(k=6, n, v[k] = (-v[k-1] * v[k-4] + v[k-2] * v[k-3]) / v[k-5]); v[n]))};

{a(n) = sign(n) * subst(elldivpol(ellinit([1, -1, 0, -1, 1]), abs(n)), x, 0) / (if(n%2, 1, 2) * (-1)^((n-1)\2) * 2^(n^2\4))}; /* Michael Somos, Feb 23 2020 */

{a(n) = my(E=ellinit([1, -1, 0, -1, 1]), z=ellpointtoz(E, [0, 1])); (-1)^(n\2) * round(ellsigma(E, n*z) / (ellsigma(E, z)^n^2 * 2^(n^2\4))) }; /* Michael Somos, Feb 25 2020 */

from gmpy2 import divexact
A242107 = [0,1,1,1,1,-1]
for n in range(6,30):
    A242107.append(int(divexact(-A242107[n-1]*A242107[n-4]+A242107[n-2]*A242107[n-3],A242107[n-5])))
print(A242107) # Chai Wah Wu, Aug 15 2014

a(n) = -(-1)^n * a(-n) for all n in Z.
0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.
0 = a(n)*a(n+7) - a(n+1)*a(n+6) - 2*a(n+2)*a(n+5) for all n in Z.
0 = a(n)*a(n+4) + a(n+1)*a(n+3) - a(n+2)*a(n+2) for all even n in Z.
0 = a(n)*a(n+4) + 2*a(n+1)*a(n+3) - a(n+2)*a(n+2) for all odd n in Z.
abs(a(n)) = A242108(n) for all n in Z.
a(2*n) = A178622(n) for all n in Z. - Michael Somos, Aug 21 2014
a(2*n-3) = A328380(n) for all n in Z. - Michael Somos, Feb 23 2020

Definition edited by Michael Somos, Feb 23 2020

A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(nx^3-(n+1)x^2+x) + sqrt((nx^3-(n+1)x^2+x)^2 - 4(x^3-x^2)((n+1)x^2-x)))/(2(x^3-x^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 6, 14, 1, 1, 2, 7, 21, 42, 1, 1, 2, 8, 30, 78, 132, 1, 1, 2, 9, 41, 136, 299, 429, 1, 1, 2, 10, 54, 222, 630, 1172, 1430, 1, 1, 2, 11, 69, 342, 1221, 2959, 4677, 4862, 1, 1, 2, 12, 86, 502, 2192, 6774, 14058, 18947, 16796, 1, 1, 2, 13, 105, 708, 3687, 14129, 37853, 67472, 77746, 58786, 1, 1, 2, 14, 126, 966, 5874, 27184

Offset: 0

Thomas Scheuerle, Oct 28 2024

The Hankel sequence transform of row n satisfies the Somos-4 recurrence c(k) = (c(k-1) * c(k-3) + n*c(k-2)^2) / c(k-4). All Somos-4 sequences which are beginning with 1, 1, 1, 1, n, ... will be covered, but the Hankel transform will start with the terms 1, n, ... in each case.

			The array begins:
  [0] 1, 1, 2,  5, 14,  42,  132,   429,   1430, ... = A000108
  [1] 1, 1, 2,  6, 21,  78,  299,  1172,   4677, ... = A254316
  [2] 1, 1, 2,  7, 30, 136,  630,  2959,  14058, ...
  [3] 1, 1, 2,  8, 41, 222, 1221,  6774,  37853, ...
  [4] 1, 1, 2,  9, 54, 342, 2192, 14129,  91494, ...
  [5] 1, 1, 2, 10, 69, 502, 3687, 27184, 201045, ...

Cf. A377442 (extension for -n), A105633 (row -1), A152172 (row -2).
Cf. A000108 (row 0), A254316 (row 1).
Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).
Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).
Cf. A371965, A377443.

T(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))

The generating function A(x) of row n satisfies: 0 = (x^3 - x^2)*A(x)^2 + (n*x^3 - (n+1)*x^2 - x)*A(x) + ((n+1)*x^2 - x).
Let d(m, n) = ( d(m-3, n)*d(m-2, n) + n)/( d(m-5, n)*d(m-4, n)*d(m-3, n)^2*d(m-2, n)^2*d(m-1, n) ) for m = even and d(m, n) = 1/( d(m-1, n)*d(m-2, n) ) for m = odd with d( < 1 , n) = 1, then the generating function of row n can be expanded as continued fractions: 1/(1 - x/(1 - d(0, n)*x/(1 - d(1, n)*x/(1 - d(2, n)*x/(...))))).
d(m, n)*d(m+1, n) is a rational solution in x of the elliptic equation y^2 = -4*x^3 + ((n+1)^2 + 8)*x^2 - 2*(n+3)*x + 1. The division polynomials for multiples of the point with x = 1, correspondent to the Hankel transform of row n in the array T(n, k).
T(n, k + 2) = Sum_{j >= 0} A377443(k, j)*n^j. This polynomial starts with A000108(k+2) + A371965(k+2)*n + ..., where A371965 is known to count peaks in the set of Catalan words of length k.

A152172 a(n) is the number of Dyck paths of semilength n without height of peaks 0 (mod 3) and height of valleys 1 (mod 3).

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 26, 59, 138, 332, 814, 2028, 5118, 13054, 33598, 87143, 227542, 597640, 1577866, 4185108, 11146570, 29798682, 79932298, 215072896, 580327122, 1569942098, 4257254850, 11569980794, 31508150890, 85968266198, 234975421554, 643317390627

Offset: 0

Jun Ma (majun(AT)math.sinica.edu.tw), Nov 27 2008

nonn

Hankel transform gives A328380(n+1). - Thomas Scheuerle, Oct 23 2024

Shu-Chung Liu, Jun Ma, Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl Math. 121 (3) (2008) 263-289.

Almost the same as A086625. - R. J. Mathar, Dec 03 2008

Cf. A000108, A328380.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      `if`(t=0 and irem(y, 3)=0, 0, b(x-1, y-1, 1))+
      `if`(t=1 and irem(y, 3)=1, 0, b(x-1, y+1, 0))))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 23 2024

CoefficientList[Series[(1+x-2x^2-Sqrt[1-2x-3x^2+4x^4])/(2(1-x)x), {x, 0, 30}], x] (* Harvey P. Dale, Apr 10 2012 *)

{a(n) = my(A, E=ellinit([1, -1, 0, -1, 1]),b=1,c=1,v=[1]); if( n<0, 0, A = O(x); for(k=1, n, v=concat(v,(1/b)*(1/c)); b=(1/b)*(1/c); c=(1-ellmul(E,[0,1],2*k+1)[1])/b; v=concat(v,c) ); for(k=1, #v,A = 1 /(1 - v[#v+1-k]*x*A));polcoeff(A, n))}; \\ Thomas Scheuerle, Oct 23 2024

{a(n) = my(A,v=[1,1,-1,-1,3]); if( n<0, 0, A = O(x); for(k=1, n+1, v=concat(v,(1/v[#v])*(1/v[#v-1])); v=concat(v,(v[#v-2]*v[#v-1]-2)/(v[#v-4]*v[#v-3]*v[#v-2]^2*v[#v-1]^2*v[#v]))); for(k=1, #v,A = 1 /(1 - v[#v+1-k]*x*A));polcoeff(A, n))} \\ Thomas Scheuerle, Oct 23 2024

G.f.: (1+x-2*x^2-sqrt(1-2*x-3*x^2+4*x^4))/(2*(1-x)*x).
(n+1)*a(n) - 2*n*a(n-1) + (7-3*n)*a(n-2) + 4*a(n-3) + 4*(n-4)*a(n-4) = 0 for n>=4. - R. J. Mathar, Aug 14 2012
G.f.: 1 - 1/G(0) where G(k) = 1 - 1/(x  + x^2/(1 + x/G(k+1) )); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 28 2012
G.f.: 1/(1-x/(1-b_{0}*x/(1-c_{0}*x/(1-b_{1}*x/(1-c_{1}*x/(...)))))), with (1-b_{n}*c_{n}) = the x-coordinate of (2*n+1) times the point [0, 1] under the group law of the elliptic curve y^2 + x*y = x^3 - x^2 - x+1. b_{n} = (1/b_{n-1})*(1/c_{n-1}) with b_{0} = 1, also c_{n} = (c_{n-1}*b_{n-1} - 2)/(b_{n}*c_{n-2}*b_{n-2}*(c_{n-1}*b_{n-1})^2) - Thomas Scheuerle, Oct 23 2024

A377442 Square array read by rising antidiagonals: T(n, k) = A377441(-n, k), an extension of A377441 into the domain of negative n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 4, 14, 1, 1, 2, 3, 9, 42, 1, 1, 2, 2, 6, 22, 132, 1, 1, 2, 1, 5, 12, 57, 429, 1, 1, 2, 0, 6, 6, 26, 154, 1430, 1, 1, 2, -1, 9, -2, 15, 59, 429, 4862, 1, 1, 2, -2, 14, -18, 24, 24, 138, 1223, 16796, 1, 1, 2, -3, 21, -48, 77, -23, 53, 332, 3550, 58786, 1, 1, 2, -4, 30, -98, 222, -226, 102, 107, 814, 10455, 208012, 1, 1, 2

Offset: 0

Thomas Scheuerle, Nov 04 2024

The main entry for this array is A377441.

			The array begins:
  [ 0] 1, 1, 2,  5, 14,  42,  132,   429,   1430, ... = A000108
  [-1] 1, 1, 2,  4,  9,  22,   57,   154,    429, ... = A105633
  [-2] 1, 1, 2,  3,  6,  12,   26,    59,    138, ... = A152172
  [-3] 1, 1, 2,  2,  5,   6,   15,    24,     53, ...
  [-4] 1, 1, 2,  1,  6,  -2,   24,   -23,    102, ...
  [-5] 1, 1, 2,  0,  9, -18,   77,  -226,    765, ...
  [-6] 1, 1, 2, -1, 14, -48,  222,  -921,   3914, ...
  [-7] 1, 1, 2, -2, 21, -98,  531, -2756,  14373, ...
Row index written as [m] is corresponding to A377441(m, k).

Cf. A377441 (The main entry for this sequence).
Cf. A105633 (row -1), A152172 (row -2).
Cf. A000108 (row 0), A254316 (row 1).
Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).
Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).
Cf. A371965, A377443.

cp's OEIS Frontend

A242107 Reduced division polynomials associated with elliptic curve y^2 + x*y = x^3 - x^2 - x + 1 and multiples of point (0, 1).

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A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(n*x^3-(n+1)*x^2+x) + sqrt((n*x^3-(n+1)*x^2+x)^2 - 4*(x^3-x^2)*((n+1)*x^2-x)))/(2*(x^3-x^2)).

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A152172 a(n) is the number of Dyck paths of semilength n without height of peaks 0 (mod 3) and height of valleys 1 (mod 3).

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A377442 Square array read by rising antidiagonals: T(n, k) = A377441(-n, k), an extension of A377441 into the domain of negative n.

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A377441 Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(nx^3-(n+1)x^2+x) + sqrt((nx^3-(n+1)x^2+x)^2 - 4(x^3-x^2)((n+1)x^2-x)))/(2(x^3-x^2)).