A000061 -id:A000061 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 22 2014

			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, ...
...

D. Shanks. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967. Math. Comput. 22, 699, 1968.

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]

Rows: A000182 (tangent numbers), A000464, A000191, A000318, A000320, A000411, A064072-A064075, ...
Columns: A000061, A000176, A000488, A000518, ...
Cf. A235605.

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 *)

Shanks gives recurrences.

More terms from Lars Blomberg, Sep 07 2015

A349271 Array A(n, k) that generalizes Euler numbers, class numbers, and tangent numbers, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 4, 8, 11, 5, 2, 4, 16, 46, 57, 16, 1, 6, 30, 128, 352, 361, 61, 2, 8, 46, 272, 1280, 3362, 2763, 272, 2, 8, 64, 522, 3522, 16384, 38528, 24611, 1385, 2, 12, 96, 904, 7970, 55744, 249856, 515086, 250737, 7936

Offset: 1

Peter Luschny, Nov 23 2021

			Seen as an array:
[1] 1,  1,   1,    2,     5,      16,       61,        272, ... [A000111]
[2] 1,  1,   3,   11,    57,     361,     2763,      24611, ... [A001586]
[3] 1,  2,   8,   46,   352,    3362,    38528,     515086, ... [A007289]
[4] 1,  4,  16,  128,  1280,   16384,   249856,    4456448, ... [A349264]
[5] 2,  4,  30,  272,  3522,   55744,  1066590,   23750912, ... [A349265]
[6] 2,  6,  46,  522,  7970,  152166,  3487246,   93241002, ... [A001587]
[7] 1,  8,  64,  904, 15872,  355688,  9493504,  296327464, ... [A349266]
[8] 2,  8,  96, 1408, 29184,  739328, 22634496,  806453248, ... [A349267]
[9] 2, 12, 126, 2160, 49410, 1415232, 48649086, 1951153920, ... [A349268]
.
Seen as a triangle:
[1] 1;
[2] 1, 1;
[3] 1, 1,  1;
[4] 1, 2,  3,   2;
[5] 2, 4,  8,  11,    5;
[6] 2, 4, 16,  46,   57,    16;
[7] 1, 6, 30, 128,  352,   361,    61;
[8] 2, 8, 46, 272, 1280,  3362,  2763,   272;
[9] 2, 8, 64, 522, 3522, 16384, 38528, 24611, 1385;

William Y. C. Chen, Neil J. Y. Fan, and Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s), Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigendum: 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]

A235605 (array generalized Euler secant numbers).
A235606 (array generalized Euler tangent numbers).
A349264 (overview generating functions).
Rows: A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
Columns: A000003 (class numbers), A000061, A000233, A000176, A000362, A000488, A000508, A000518.
Cf. A349263 (main diagonal).

A000176 Generalized tangent numbers d_(n,2).

Original entry on oeis.org

2, 11, 46, 128, 272, 522, 904, 1408, 2160, 3154, 4306, 5888, 7888, 10012, 12888, 16384, 19680, 24354, 29866, 34816, 41888, 49778, 56744, 66816, 78000, 87358, 100602, 115712, 128112, 145804, 165712, 180224, 203040, 228964, 246932, 276480

Offset: 1

N. J. A. Sloane

nonn

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

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).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 21 (1967), 689-694; 22 (1968), 699.

Cf. A000061 for d_(n,1), A000488 for d_(n,3), A000518 for d_(n,4).

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 03 2000

A000488 Generalized tangent numbers d_(n,3).

Original entry on oeis.org

16, 361, 3362, 16384, 55744, 152166, 355688, 739328, 1415232, 2529614, 4261454, 6885376, 10708160, 16054580, 23494584, 33554432, 46698624, 64037790, 86342918, 114163712, 149518720, 193356526, 246232840, 311635968, 390600000

Offset: 1

N. J. A. Sloane

nonn

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).

Lars Blomberg, Table of n, a(n) for n = 1..1000
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699

Cf. A000061, A000176, A000518.

amax = 25; km0 = 10; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; d[1, n_, km_] := 2 (2 n - 1)! L[-1, 2 n, km] (2/Pi)^(2 n) // Round; d[a_ /; a > 1, n_, km_] := (2 n - 1)! L[-a, 2 n, km] (2 a/Pi )^(2 n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[a, 3, km], {a, 1, amax}]; dd[km0]; dd[km = 2 km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000488 = dd[km] (* Jean-François Alcover, Feb 08 2016 *)

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 03 2000

A000518 Generalized tangent numbers d_(n,4).

Original entry on oeis.org

272, 24611, 515086, 4456448, 23750912, 93241002, 296327464, 806453248, 1951153920, 4300685074, 8787223186, 16878338048, 30768878848, 53624926972, 89982082488, 146028888064, 230022888960, 353194774434, 529896144586

Offset: 1

N. J. A. Sloane

nonn

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).

Lars Blomberg, Table of n, a(n) for n = 1..1000
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699

Cf. A000061 for d_(n,1), A000176 for d_(n,2), A000488 for d_(n,3).

amax = 20; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[ JacobiSymbol[ -a, 2 k + 1]/(2 k + 1)^s, {k, 0, km}]; d[1, n_, km_] := 2 (2 n - 1)! L[-1, 2 n, km] (2/Pi)^(2 n) // Round; d[a_ /; a > 1, n_, km_] := (2 n - 1)! L[-a, 2 n, km] (2 a/Pi)^(2 n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[a, 4, km], {a, 1, amax}]; dd[km0]; dd[km = 2 km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000518 = dd[km] (* Jean-François Alcover, Feb 09 2016 *)

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 03 2000

cp's OEIS Frontend

A235606 Shanks's array d_{a,n} (a >= 1, n >= 1) that generalizes the tangent numbers, read by antidiagonals upwards.

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A349271 Array A(n, k) that generalizes Euler numbers, class numbers, and tangent numbers, read by ascending antidiagonals.

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A000176 Generalized tangent numbers d_(n,2).

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A000488 Generalized tangent numbers d_(n,3).

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A000518 Generalized tangent numbers d_(n,4).

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