OFFSET
0,3
COMMENTS
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
.
FORMULA
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.
EXAMPLE
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
PROG
(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);
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Thomas Scheuerle, Nov 17 2024
STATUS
approved