A000508 Generalized class numbers c_(n,3).

61, 2763, 38528, 249856, 1066590, 3487246, 9493504, 22634496, 48649086, 96448478, 179369856, 315621376, 530788622, 860061996, 1346126848, 2046820352, 3038120316, 4403100222, 6254596992, 8737505280, 11992903772

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Let L_a(s) = Sum_{k>=0} (-a|2k+1) /(2k+1)^s be a Dirichlet series, where (-a|2k+1) is the Jacobi symbol. Then the c_(a,n) are defined by L_a(2n+1) = (Pi/(2a))^(2n+1)*sqrt(a)*c_(a,n)/(2n)! for n=0,1,2,..., a=1,2,3,...

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Matthew House, Table of n, a(n) for n = 1..10000
• D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 1967 6890694.
• D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
• D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Crossrefs

Column 3 of A235605.

Cf. A000003, A000233, A000362.

Programs

Mathematica

amax = 25; km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[ -a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; c[1, n_, km_] := 2 (2 n)! L[1, 2 n + 1, km] (2/Pi)^(2 n + 1) // Round; c[a_ /; a > 1, n_, km_] := (2 n)! L[a, 2 n + 1, km] (2 a/Pi)^(2 n + 1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[ c[a, 3, km], {a, 1, amax} ]; cc[km0]; cc[km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000508 = cc[km] (* Jean-François Alcover, Feb 09 2016 *)
Table[rowA235605[n, 3][[4]], {n, 50}] (* see A235605 *) (* Matthew House, Oct 05 2024 *)

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 02 2000

Name clarified by James C. McMahon, Nov 30 2023