A235606 Shanks's array d_{a,n} (a >= 1, n >= 1) that generalizes the tangent numbers, read by antidiagonals upwards.
1, 1, 2, 2, 11, 16, 4, 46, 361, 272, 4, 128, 3362, 24611, 7936, 6, 272, 16384, 515086, 2873041, 353792, 8, 522, 55744, 4456448, 135274562, 512343611, 22368256, 8, 904, 152166, 23750912, 2080374784, 54276473326, 129570724921, 1903757312, 12, 1408, 355688
Offset: 1
Examples
The array begins: A000182: 1, 2, 16, 272, 7936, 353792, ... A000464: 1, 11, 361, 24611, 2873041, 512343611, ... A000191: 2, 46, 3362, 515086, 135274562, 54276473326, ... A000318: 4,128, 16384, 4456448, 2080374784, 1483911200768, ... A000320: 4,272, 55744, 23750912, 17328937984, 19313964388352, ... A000411: 6,522,152166, 93241002, 97949265606,157201459863882, ... A064072: 8,904,355688,296327464,423645846728,925434038426824, ... ...
References
- D. Shanks. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967. Math. Comput. 22, 699, 1968.
Links
- Lars Blomberg, Table of n, a(n) for n = 1..5050
- D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
Crossrefs
Programs
-
Mathematica
amax = nmax = 10; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[ -a, 2k+1]/(2k+1)^s, {k, 0, km}]; d[1, n_, km_] := 2(2n-1)! L[-1, 2n, km] (2/Pi)^(2n) // Round; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/ Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[a, n, km], {a, 1, amax}, {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2km]]; A235606 = dd[km]; Table[A235606[[ a-n+1, n]], {a, 1, amax}, {n, 1, a}] // Flatten (* Jean-François Alcover, Feb 05 2016 *) dds[b_, nm_] := With[{ns = Range[nm]}, (-1)^(ns - 1) If[Mod[b, 4] == 1, Sum[JacobiSymbol[k, b] (b - 4 k)^(2 ns - 1), {k, 1, (b - 1)/2}], Sum[JacobiSymbol[b, 2 k + 1] (b - (2 k + 1))^(2 ns - 1), {k, 0, (b - 2)/2}]]]; dsfs[1, nm_] := dsfs[1, nm] = (2 Range[nm] - 1)! CoefficientList[Series[Tan[x], {x, 0, 2 nm - 1}]/x, x^2]; dsfs[b_, nm_] := dsfs[b, nm] = Fold[Function[{ds, dd}, Append[ds, dd - Sum[ds[[-i]] (-b^2)^i Binomial[2 Length[ds] + 1, 2 i], {i, Length[ds]}]]], {}, dds[b, nm]]; rowA235606[a_, nm_] := With[{facs = FactorInteger[a], ns = Range[nm]}, With[{b = Times @@ (#^Mod[#2, 2] &) @@@ facs}, If[a == b, dsfs[b, nm], If[b == 1, 1/2, 1] dsfs[b, nm] Sqrt[a/b]^(4 ns - 1) Times @@ Cases[facs, {p_, e_} /; p > 2 && e > 1 :> 1 - JacobiSymbol[b, p]/p^(2 ns)]]]]; arr = Table[rowA235606[a, 10], {a, 10}]; Flatten[Table[arr[[r - n + 1, n]], {r, Length[arr]}, {n, r}]] (* Matthew House, Oct 30 2024 *)
Formula
Shanks gives recurrences.
Extensions
More terms from Lars Blomberg, Sep 07 2015