A000508 -id:A000508 - cp's OEIS frontend

A235605 Shanks's array c_{a,n} (a >= 1, n >= 0) that generalizes Euler and class numbers, read by antidiagonals upwards.

1, 1, 1, 1, 3, 5, 1, 8, 57, 61, 2, 16, 352, 2763, 1385, 2, 30, 1280, 38528, 250737, 50521, 1, 46, 3522, 249856, 7869952, 36581523, 2702765, 2, 64, 7970, 1066590, 90767360, 2583554048, 7828053417, 199360981, 2, 96, 15872, 3487246, 604935042, 52975108096

Offset: 1

N. J. A. Sloane, Jan 22 2014

			The array begins:
A000364: 1, 1,    5,     61,       1385,         50521,          2702765,..
A000281: 1, 3,   57,   2763,     250737,      36581523,       7828053417,..
A000436: 1, 8,  352,  38528,    7869952,    2583554048,    1243925143552,..
A000490: 1,16, 1280, 249856,   90767360,   52975108096,   45344872202240,..
A000187: 2,30, 3522,1066590,  604935042,  551609685150,  737740947722562,..
A000192: 2,46, 7970,3487246, 2849229890, 3741386059246, 7205584123783010,..
A064068: 1,64,15872,9493504,10562158592,18878667833344,49488442978598912,..
...

Matthew House, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals; first 100 antidiagonals from Lars Blomberg)
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]

Rows: A000364 (Euler numbers), A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070, A064071, ...
Columns: A000003 (class numbers), A000233, A000362, A000508, ...
Cf. A235606.

amax = 10; nmax = amax-1; km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2k+1]/(2k+1)^s, {k, 0, km}]; c[1, n_, km_] := 2(2n)! L[1, 2n+1, km] (2/Pi)^(2n+1) // Round; c[a_ /; a>1, n_, km_] := (2n)! L[a, 2n+1, km] (2a/Pi)^(2n+1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[ c[a, n, km], {a, 1, amax}, {n, 0, nmax}]; cc[km0]; cc[km = 2km0]; While[ cc[km] != cc[km/2, km = 2km]]; A235605[a_, n_] := cc[km][[a, n+1 ]]; Table[ A235605[ a-n, n], {a, 1, amax}, {n, 0, a-1}] // Flatten (* Jean-François Alcover, Feb 05 2016 *)
ccs[b_, nm_] := With[{ns = Range[0, nm]}, (-1)^ns If[Mod[b, 4] == 3, Sum[JacobiSymbol[k, b] (b - 4 k)^(2 ns), {k, 1, (b - 1)/2}], Sum[JacobiSymbol[-b, 2 k + 1] (b - (2 k + 1))^(2 ns), {k, 0, (b - 2)/2}]]];
csfs[1, nm_] := csfs[1, nm] = (2 Range[0, nm])! CoefficientList[Series[Sec[x], {x, 0, 2 nm}], x^2];
csfs[b_, nm_] := csfs[b, nm] = Fold[Function[{cs, cc}, Append[cs, cc - Sum[cs[[-i]] (-b^2)^i Binomial[2 Length[cs], 2 i], {i, Length[cs]}]]], {}, ccs[b, nm]];
rowA235605[a_, nm_] := With[{facs = FactorInteger[a], ns = Range[0, nm]}, With[{b = Times @@ (#^Mod[#2, 2] &) @@@ facs}, If[a == b, csfs[b, nm], If[b == 1, 1/2, 1] csfs[b, nm] Sqrt[a/b]^(4 ns + 1) Times @@ Cases[facs, {p_, e_} /; p > 2 && e > 1 :> 1 - JacobiSymbol[-b, p]/p^(2 ns + 1)]]]];
arr = Table[rowA235605[a, 10], {a, 10}];
Flatten[Table[arr[[r - n + 1, n + 1]], {r, 0, Length[arr] - 1}, {n, 0, r}]] (* Matthew House, Sep 07 2024 *)

Shanks gives recurrences.

a(27) removed, a(29)-a(42) added, and typo in name corrected by Lars Blomberg, Sep 10 2015

Offset corrected by Andrew Howroyd, Oct 25 2024

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).

A000233 Generalized class numbers c_(n,1).

Original entry on oeis.org

1, 3, 8, 16, 30, 46, 64, 96, 126, 158, 216, 256, 302, 396, 448, 512, 636, 702, 792, 960, 1052, 1118, 1344, 1472, 1550, 1866, 1944, 2048, 2442, 2540, 2688, 3072, 3212, 3388, 3888, 4032, 4094, 4746, 4928, 5056, 5832, 5852, 5976, 6912, 7020, 7180, 8064, 8192

Offset: 1

N. J. A. Sloane

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,...

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).

Matthew House, Table of n, a(n) for n = 1..10000
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]

Cf. A000508, A000362.

amax = 50; nmax = 1; 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, n, km], {a, 1, amax}, {n, 0, nmax}]; cc[km0]; cc[ km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000233 = cc[km][[All, 2]] (* Jean-François Alcover, Feb 06 2016 *)
Table[rowA235605[n, 1][[2]], {n, 50}] (* see A235605 *) (* Matthew House, Oct 05 2024 *)

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

Name clarified by James C. McMahon, Nov 30 2023

A000362 Generalized class numbers c_(n,2).

Original entry on oeis.org

5, 57, 352, 1280, 3522, 7970, 15872, 29184, 49410, 79042, 122400, 180224, 257314, 362340, 492032, 655360, 867588, 1117314, 1420320, 1803264, 2237380, 2745154, 3380736, 4080640, 4881250, 5874150, 6928416, 8126464, 9600870, 11133604

Offset: 1

N. J. A. Sloane

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).

Matthew House, 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. 22 (1968), 699
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Cf. A000233, A000508.

amax = 30;   km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k+1]/(2k+1)^s, {k, 0, km}]; c[1, n_, km_] := 2(2n)! L[1, 2n+1, km] (2 / Pi)^(2n+1) // Round; c[a_ /; a>1, n_, km_] := (2n)! L[a, 2n+1, km] (2a / Pi)^(2n+1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[c[a, n, km], {a, 1, amax}, {n, 0, nmax}]; cc[km0]; cc[km = 2km0]; While[cc[km] != cc[km/2, km = 2km]]; A000362[a_] := cc[km][[a, 3]]; Table[A000362[a], {a, 1, amax} ] (* Jean-François Alcover, Feb 08 2016 *)
Table[rowA235605[n, 2][[3]], {n, 50}] (* see A235605 *) (* Matthew House, Oct 05 2024 *)

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

cp's OEIS Frontend

A235605 Shanks's array c_{a,n} (a >= 1, n >= 0) that generalizes Euler and class 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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A000233 Generalized class numbers c_(n,1).

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A000362 Generalized class numbers c_(n,2).

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