A235605 -id:A235605 - cp's OEIS frontend

A000362 Generalized class numbers c_(n,2).

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

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, 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

A000508 Generalized class numbers c_(n,3).

Original entry on oeis.org

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

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 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]

Column 3 of A235605.

Cf. A000003, A000233, A000362.

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 *)

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

Name clarified by James C. McMahon, Nov 30 2023

A064070 Generalized Euler number c(9,n).

Original entry on oeis.org

2, 126, 49410, 48649086, 89434106370, 264235243691646, 1145011717430672130, 6841110155700330881406, 53899295662946509072626690, 541439307193573593050370186366, 6754273504043546592593642328610050, 102439130403410639137159601119206854526

Offset: 0

Eric W. Weisstein, Aug 31 2001

Matthew House, Table of n, a(n) for n = 0..174
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.
Eric Weisstein's World of Mathematics, Euler Number.

Row 9 of A235605.

Cf. A064074, A349268, A349264.

egf := sec(9*x)*2*cos(3*x)^2: ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..10); # Peter Luschny, Nov 21 2021

Range[0, 22, 2]! CoefficientList[Series[2 Sec[9 x] Cos[3 x]^2, {x, 0, 22}], x^2] (* Matthew House, Oct 27 2024 *)

a(n) = (2*n)!*[x^(2*n)](sec(9*x)*2*cos(3*x)^2). - Peter Luschny, Nov 21 2021

A064071 Generalized Euler number c(10,n).

Original entry on oeis.org

2, 158, 79042, 96448478, 218987626882, 798807132800798, 4273446686680147522, 31521738825258026305118, 306606728264687314446344962, 3802448636519829800140388609438

Offset: 0

Eric W. Weisstein, Aug 31 2001

Matthew House, Table of n, a(n) for n = 0..171
Eric Weisstein's World of Mathematics, Euler Number.
Daniel Shanks, Generalized Euler and class numbers, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 689-694; Corrigenda, ibid., Vol. 22, No. 103 (1968), p. 699.

Row 10 of A235605.

Range[0, 18, 2]! CoefficientList[Series[Sec[10 x] (Cos[x] + Cos[3 x] - Cos[7 x] + Cos[9 x]), {x, 0, 18}], x^2] (* Matthew House, Oct 27 2024 *)

a(n) = (2*n)!*[x^(2*n)](sec(10*x)*(cos(x) + cos(3*x) - cos(7*x) + cos(9*x))). - Matthew House, Oct 27 2024

A262144 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} d(n,i+1)*x^i/i ) for n >= 1, where d(n,k) is Shanks's array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 2, 1, 11, 10, 1, 46, 241, 108, 1, 128, 2739, 10411, 2214, 1, 272, 16384, 265244, 836321, 75708, 1, 522, 64964, 2883584, 45094565, 112567243, 3895236, 1, 904, 212325, 18852096, 822083584, 12975204810, 22949214033

Offset: 1

Peter Bala, Sep 18 2015

Shanks's array d(n,k) n >= 1, k >= 1, is A235606.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 1, 2, ... and for each n >= 1, the expansion of exp( Sum_{i >= 1} d(n,i + r)*x^i/i ) has integer coefficients. This is the case r = 1.
For the similarly defined array associated with Shanks' c(n,k) array see A262143.

			The triangular array begins
1
1   2
1  11     10
1  46    241      108
1 128   2739    10411      2214
1 272  16384   265244    836321       75708
1 522  64964  2883584  45094565   112567243     3895236
1 904 212325 18852096 822083584 12975204810 22949214033 ...
The square array begins (row indexing n starts at 1)
1, 2, 10, 108, 2214, 75708, 3895236, 280356120, 26824493574, ...
1, 11, 241, 10411, 836321, 112567243, 22949214033, 6571897714923, 2507281057330113, ...
1, 46, 2739, 265244, 45094565, 12975204810, 5772785327575, 3656385436507960, 3107332328608143945, ...
1, 128, 16384, 2883584, 822083584, 395136991232, 300338473074688, 330739694704787456, 493338658405976375296, ...
1, 272, 64864, 18852096, 8133183744, 5766226378752, 6562478680375296, 11019751545852395520, 25333348417380699340800, ...
1, 522, 212325, 94501768, 57064909374, 54459242196516, 84430282319806062, 197625548666434041000, 642556291067409622713543, ...
1, 904, 586452, 382674008, 311514279098, 379982635729752, 753288329161251844, 2308779464340711480136, 10003494921382094286802995, ...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia , The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
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. A000182 (d(1,n)), A000464 (d(2,n)), A000191 (d(3,n)), A000318 (d(4,n)), A000320 (d(5,n)), A000411 (d(6,n)), A064072 (d(7,n)), A235605, A235606, A262143, A262145 (row 1 of square array).

A262143 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 8, 33, 23, 1, 16, 208, 1011, 371, 1, 30, 768, 14336, 65985, 10515, 1, 46, 2211, 94208, 2091520, 7536099, 461869, 1, 64, 5043, 412860, 24313856, 535261184, 1329205857, 28969177, 1, 96, 9984, 1361948, 164276421, 11025776640, 211966861312, 334169853267, 2454072147

Offset: 1

Peter Bala, Sep 13 2015

Shanks' array c(n,k) n >= 1, k >= 0, is A235605.
We conjecture that the entries of the present array are all integers. More generally, we conjecture that for r = 0,1,2,... and for each n >= 1, the expansion of exp( Sum_{i >= 1} c(n,i + r)*x^i/i ) has integer coefficients. The case n = 1 was conjectured by Hanna in A255895.
For the similarly defined array associated with Shanks' d(n,k) array see A262144.

			The square array begins (row indexing n starts at 1)
1  1    3      23        371         10515           461869 ..
1  3   33    1011      65985       7536099       1329205857 ..
1  8  208   14336    2091520     535261184     211966861312 ..
1 16  768   94208   24313856   11025776640    7748875976704 ..
1 30 2211  412860  164276421  115699670490  126686112278631 ..
1 46 5043 1361948  778121381  787337024970 1239870854518999 ..
1 64 9984 3716096 2891509760 3978693525504 8522989918683136 ..
...
Array as a triangle
1
1  1
1  3    3
1  8   33      23
1 16  208    1011      371
1 30  768   14336    65985        10515
1 46 2211   94208  2091520      7536099       461869
1 64 5043  412860  24313856   535261184   1329205857 28969177
1 96 9984 1361948 164276421 11025776640 211966861312 ...
...

P. Bala, Notes on logarithmic differentiation, the binomial transform and series reversion
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s) Mathematics of Computation, Vol. 81, No. 278, April 2012.
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 (column 1), A000364 (c(1,n)), A000281 (c(2,n)), A000436 (c(3,n)), A000490 (c(4,n)), A000187 (c(5,n)), A000192 (c(6,n)), A064068 (c(7,n)), A235605, A235606, A255881, A255895, A262144, A262145.

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A000362 Generalized class numbers c_(n,2).

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A000508 Generalized class numbers c_(n,3).

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A064070 Generalized Euler number c(9,n).

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A064071 Generalized Euler number c(10,n).

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A262144 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} d(n,i+1)*x^i/i ) for n >= 1, where d(n,k) is Shanks's array of generalized Euler and class numbers.

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A262143 Square array read by antidiagonals upwards: the n-th row o.g.f. is exp( Sum_{i >= 1} c(n,i)*x^i/i ) for n >= 1, where c(n,k) is Shanks' array of generalized Euler and class numbers.

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