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

1,3

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]

Formula

Shanks gives recurrences.

Example

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, ...
...

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 *)

Crossrefs

Rows: A000182 (tangent numbers), A000464, A000191, A000318, A000320, A000411, A064072-A064075, ...

Columns: A000061, A000176, A000488, A000518, ...

Cf. A235605.

Sequence in context: A121871 A194638 A327870 * A175202 A187430 A151365

Adjacent sequences: A235603 A235604 A235605 * A235607 A235608 A235609

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 22 2014

Extensions

More terms from Lars Blomberg, Sep 07 2015

Status

approved