0,3

Unfortunately, if m is a fraction m = b/c, this triangle can only be used for those coefficients where c and (n+1) are coprime. This is not only because the modulo operation is undefined otherwise, but also because rows of the triangle where (n+1) divides c contain these coefficients with the wrong sign.

The parametrization model for elliptic equations defined by -4*x^3 + ((m+1)^2 + 8)*x^2 - 2*(m+3)*x + 1 - y^2 is also used in A377441. From its relation to Somos-4 sequences, it is known that there is at least one generator point of infinite order if m is an integer > 0 or < -1. If we assume the Birch and Swinnerton-Dyer conjecture to be true, then we expect the associated L-function L(E, s) to be zero at s = 1 for such m.

The relation of m to the J-invariant is given by J(m) = (m^12 + 12*m^11 + 114*m^10 + 628*m^9 + 2823*m^8 + 8184*m^7 + 19036*m^6 + 24552*m^5 + 25407*m^4 + 16956*m^3 + 9234*m^2 + 2916*m + 729)/(m^5 + 4*m^4 + 23*m^3 + 9*m^2) for rational m.

The row sums of the triangle show some connection to the Dedekind psi function (A001615), but will deviate for at least many nonsquarefree n+1.

A short table which shows the Cremona label which corresponds to the L-series obtained for some rational m:

.

m | label

-------------

-5 655a1

-4 166a1

-3 153a1

-2 58a1

-1 11a3

-1/2 26b1

-1/3 141a1

1 37a1

2 158b1

3 423g1

4 458a1

5 1745b1

.

T(n, n) = A006571(n), case m =-1. Also the expansion of (eta(q) * eta(q^11))^2 in powers of q.

T(n, 1) = A007653(n), case m = 1.

T(2*n, n) = A251913(2*n+1), case m = -1/2. See first comment.

Let p be an odd prime with good reduction, then T(p-1, k) is odd iff -4*x^3 + ((k+1)^2 + 8)*x^2 - 2*(k+3)*x + 1 == 0 (mod p) has no solution.

The triangle T(n, k) begins:

q^(n+1) 0, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 sum A001615

--------------------------------------------------------------------

[q^1] 1 1 1

[q^2] -1,-2 -3 3

[q^3] 0,-3,-1 -4 4

[q^4] 1, 2, 1, 2 6 6

[q^5] -1,-2,-1,-3, 1 6 6

[q^6] 0, 6, 1, 0, 3, 2 12 12

[q^7] 1,-1,-3, 1,-2,-2,-2 -8 8

[q^8] -1, 0,-1, 0,-1, 0,-1, 0 -4 12 <- not equal

[q^9] 0, 6,-2, 0, 6,-2, 0, 6,-2 12 12

[q^10] 1, 4, 1, 6,-1, 2, 2, 2, 3,-2 18 18

[q^11] -1,-5, 4, 3, 1,-2,-4,-5,-3,-1, 1 12 12

[q^12] 0,-6,-1, 0,-3,-2, 0,-6,-1, 0,-3,-2 -24 24

[q^13] 1,-2,-7, 0, 2,-2,-1, 0,-5,-2,-5, 3, 4 -14 14

[q^14] -1, 2, 3,-2, 2, 4, 2,-2, 1, 6,-1, 4, 2, 4 24 24

[q^15] 0, 6, 1, 0,-3, 1, 0, 3, 3, 0, 3, 2, 0, 9,-1 24 24

[q^16] 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4 24 24

(PARI)

T(n, k) = ellak(ellinit(ellfromeqn(-4*x^3 + ((k+n+2)^2 + 8)*x^2 - 2*(k+n+4)*x + 1 - y^2)), n+1);

Cf. A001615, A251913, A377441.

Cf. A006571 (main diagonal), A007653 (column 1).

Sequence in context: A357624 A353923 A246370 * A362557 A047983 A070812

Adjacent sequences: A378122 A378123 A378124 * A378126 A378127 A378128

sign,tabl

Thomas Scheuerle, Nov 17 2024

approved