This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378125 #34 Dec 08 2024 04:00:44 %S A378125 1,-1,-2,0,-3,-1,1,2,1,2,-1,-2,-1,-3,1,0,6,1,0,3,2,1,-1,-3,1,-2,-2,-2, %T A378125 -1,0,-1,0,-1,0,-1,0,0,6,-2,0,6,-2,0,6,-2,1,4,1,6,-1,2,2,2,3,-2,-1,-5, %U A378125 4,3,1,-2,-4,-5,-3,-1,1,0,-6,-1,0,-3,-2,0,-6,-1,0,-3,-2,1,-2,-7,0,2,-2,-1,0,-5,-2,-5,3,4,-1,2,3,-2,2,4,2,-2,1,6,-1,4,2,4 %N A378125 Triangle T(n, k) read by rows. Let m be a nonzero rational number then T(n, m mod (n+1)) is the n-th coefficient in the Hasse-Weil L-series (q^(n+1) in the q-expansion) associated to the elliptic equation -4*x^3 + ((m+1)^2 + 8)*x^2 - 2*(m+3)*x + 1 - y^2 = 0. %C A378125 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. %C A378125 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. %C A378125 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. %C A378125 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. %C A378125 A short table which shows the Cremona label which corresponds to the L-series obtained for some rational m: %C A378125 . %C A378125 m | label %C A378125 ------------- %C A378125 -5 655a1 %C A378125 -4 166a1 %C A378125 -3 153a1 %C A378125 -2 58a1 %C A378125 -1 11a3 %C A378125 -1/2 26b1 %C A378125 -1/3 141a1 %C A378125 1 37a1 %C A378125 2 158b1 %C A378125 3 423g1 %C A378125 4 458a1 %C A378125 5 1745b1 %C A378125 . %F A378125 T(n, n) = A006571(n), case m =-1. Also the expansion of (eta(q) * eta(q^11))^2 in powers of q. %F A378125 T(n, 1) = A007653(n), case m = 1. %F A378125 T(2*n, n) = A251913(2*n+1), case m = -1/2. See first comment. %F A378125 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. %e A378125 The triangle T(n, k) begins: %e A378125 q^(n+1) 0, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 sum A001615 %e A378125 -------------------------------------------------------------------- %e A378125 [q^1] 1 1 1 %e A378125 [q^2] -1,-2 -3 3 %e A378125 [q^3] 0,-3,-1 -4 4 %e A378125 [q^4] 1, 2, 1, 2 6 6 %e A378125 [q^5] -1,-2,-1,-3, 1 6 6 %e A378125 [q^6] 0, 6, 1, 0, 3, 2 12 12 %e A378125 [q^7] 1,-1,-3, 1,-2,-2,-2 -8 8 %e A378125 [q^8] -1, 0,-1, 0,-1, 0,-1, 0 -4 12 <- not equal %e A378125 [q^9] 0, 6,-2, 0, 6,-2, 0, 6,-2 12 12 %e A378125 [q^10] 1, 4, 1, 6,-1, 2, 2, 2, 3,-2 18 18 %e A378125 [q^11] -1,-5, 4, 3, 1,-2,-4,-5,-3,-1, 1 12 12 %e A378125 [q^12] 0,-6,-1, 0,-3,-2, 0,-6,-1, 0,-3,-2 -24 24 %e A378125 [q^13] 1,-2,-7, 0, 2,-2,-1, 0,-5,-2,-5, 3, 4 -14 14 %e A378125 [q^14] -1, 2, 3,-2, 2, 4, 2,-2, 1, 6,-1, 4, 2, 4 24 24 %e A378125 [q^15] 0, 6, 1, 0,-3, 1, 0, 3, 3, 0, 3, 2, 0, 9,-1 24 24 %e A378125 [q^16] 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4, 1,-4 24 24 %o A378125 (PARI) %o A378125 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); %Y A378125 Cf. A001615, A251913, A377441. %Y A378125 Cf. A006571 (main diagonal), A007653 (column 1). %K A378125 sign,tabl %O A378125 0,3 %A A378125 _Thomas Scheuerle_, Nov 17 2024