A349264 -id:A349264 - cp's OEIS frontend

A000318 Generalized tangent numbers d(4,n).

4, 128, 16384, 4456448, 2080374784, 1483911200768, 1501108249821184, 2044143848640217088, 3605459138582973251584, 7995891855149741436305408, 21776918737280678860353961984, 71454103701490016776039304265728, 278008871543597996197497752082448384

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).

T. D. Noe, Table of n, a(n) for n = 1..100
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]
Eric Weisstein's World of Mathematics, Padé approximants.

Cf. A000182, A000191, A000490, A349264.

egf := sec(4*x)*sin(4*x): ser := series(egf, x, 26):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..12); # Peter Luschny, Nov 21 2021

nn = 30; t = Rest@Union[Range[0, nn - 1]! CoefficientList[Series[Tan[x], {x, 0, nn}], x]]; t2 = t*2^Range[2, 2*nn, 4] (* T. D. Noe, Jun 19 2012 *)

a(n) = 2^(4n-2) * A000182(n).
The g.f. has the following continued fraction expansion: g.f. = [4, b(0), c(0), b(1), c(1), b(2), c(2), ...] where b(n) = (Sum_{k=0..n} 1/(2*k+1))^2 / (128*(n+1)*x), c(n) = -4/((2*n+3)*(Sum_{k=0..n} 1/(2*k+1))*(Sum_{k=0..n+1} 1/(2*k+1))) and each convergent of this continued fraction is a Padé approximant of the Maclaurin series Sum_{k>=1} a(n)*x^(n-1). - Thomas Baruchel, Oct 19 2005
a(n) = (2*n-1)!*[x^(2*n-1)](sec(4*x)*sin(4*x)). - Peter Luschny, Nov 21 2021

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

A000490 Generalized Euler numbers c(4,n).

Original entry on oeis.org

1, 16, 1280, 249856, 90767360, 52975108096, 45344872202240, 53515555843342336, 83285910482761809920, 165262072909347030040576, 407227428060372417275494400, 1219998300294918683087199010816, 4366953142363907901751614431559680, 18406538229888710811704852978971181056

Offset: 0

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 = 0..194
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]

Row 4 of A235605.

Cf. A000187, A000436, A000318, A349264.

egf := sec(4*x): ser := series(egf, x, 26):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

a0 = 4; nmax = 20; km0 = nmax; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2*k+1]/(2*k+1)^s, {k, 0, km}]; c[a_, n_, km_] := 2^(2*n +1)*Pi^(-(2*n)-1)*(2*n)!*a^(2*n+1/2)*L[a, 2*n+1, km] // Round; cc[km_] := cc[km] = Table[c[a0, n, km], {n, 0, nmax}]; cc[km0]; cc[km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000490 = cc[km] (* Jean-François Alcover, Feb 05 2016 *)
Range[0, 26, 2]! CoefficientList[Series[Sec[4 x], {x, 0, 26}], x^2] (* Matthew House, Oct 05 2024 *)

a(n) = A000364(n)*16^n. - Philippe Deléham, Oct 27 2006

a(n) = (2*n)!*[x^(2*n)](sec(4*x)). - Peter Luschny, Nov 21 2021

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

A001587 Generalized Euler numbers.

Original entry on oeis.org

2, 6, 46, 522, 7970, 152166, 3487246, 93241002, 2849229890, 97949265606, 3741386059246, 157201459863882, 7205584123783010, 357802951084619046, 19133892392367261646, 1096291279711115037162, 67000387673723462963330, 4350684698032741048452486, 299131045427247559446422446

Offset: 0

N. J. A. Sloane

nonn

These numbers are related to the values at negative integers of the L-functions for two primitive Dirichlet characters of conductor 24. - F. Chapoton, Oct 05 2020

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 = 0..199
LMFDB, character 24.5
LMFDB, character 24.11
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]

Bisections are A000192 and A000411. Overview in A349264.

Similar sequences: A000111, A225147.

egf := sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)): ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # Peter Luschny, Nov 21 2021

t = PowerSeriesRing(QQ, 't').gen()
f = 2 * (sin(3 * t) + cos(3 * t)) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list() # F. Chapoton, Oct 06 2020

E.g.f.: 2 (sin(3 x) + cos(3 x)) / (2 cos(4 x) - 1). - F. Chapoton, Oct 06 2020
a(n) ~ 2^(2*n + 2) * 3^(n + 1/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Nov 05 2021
a(n) = n!*[x^n](sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x))). - Peter Luschny, Nov 21 2021

a(11)-a(14) from Lars Blomberg, Sep 10 2015

A064068 Generalized Euler number c(7,n).

Original entry on oeis.org

1, 64, 15872, 9493504, 10562158592, 18878667833344, 49488442978598912, 178867627497727197184, 852509723495811705208832, 5180564635674867885905281024, 39094622102339738427522497380352, 358686739310560735577543742129700864, 3931974790759726002374736527410407145472

Offset: 0

Eric W. Weisstein, Aug 31 2001

Matthew House, Table of n, a(n) for n = 0..180
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 7 of A235605.

Cf. A064072, A349266, A349264.

egf := sec(7*x)*(cos(x) + cos(3*x) - cos(5*x)): ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

Range[0, 24, 2]! CoefficientList[Series[Sec[7 x] (Cos[x] + Cos[3 x] - Cos[5 x]), {x, 0, 24}], x^2] (* Matthew House, Oct 25 2024 *)

a(n) = (2*n)!*[x^(2*n)](sec(7*x)*(cos(x) + cos(3*x) - cos(5*x))). - Peter Luschny, Nov 21 2021

A064069 Generalized Euler number c(8,n).

Original entry on oeis.org

2, 96, 29184, 22634496, 32864600064, 76717014122496, 262665886073094144, 1239981021847665770496, 7719096548270543600615424, 61267211781784116552580202496, 603881788505747521507846892027904, 7236592671961544936200760521440362496, 103612803724706836868168667250308188995584

Offset: 0

Eric W. Weisstein, Aug 31 2001

Matthew House, Table of n, a(n) for n = 0..176
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 8 of A235605.

Cf. A000281, A064073, A349267, A349264.

egf := sec(8*x)*2*cos(4*x): ser := series(egf, x, 24):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

Range[0, 24, 2]! CoefficientList[Series[2 Sec[8 x] Cos[4 x], {x, 0, 24}], x^2] (* Matthew House, Oct 25 2024 *)

a(n) = 2^(4n+5) * A000281(n).

a(n) = (2*n)!*[x^(2*n)](sec(8*x)*2*cos(4*x)). - Peter Luschny, Nov 21 2021

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

A064073 Generalized tangent number d(8,n).

Original entry on oeis.org

8, 1408, 739328, 806453248, 1506300919808, 4297849713983488, 17390688314209599488, 94727563504456856240128, 668321603392783694711226368, 5928595592752632717848942215168, 64586438563324327821773422563688448, 847680268223550650928681687352090820608

Offset: 1

Eric W. Weisstein, Aug 31 2001

Lars Blomberg, Table of n, a(n) for n = 1..177
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, Tangent Number.

Cf. A000464, A064073, A064069, A349267, A349264.

egf := sec(8*x)*2*sin(4*x): ser := series(egf, x, 24):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..10); # Peter Luschny, Nov 21 2021

E.g.f.: Sum_{k>0} a(k)x^(2k-1)/(2k-1)! = 2*sin(4x)/cos(8x).
a(n) = 2^(4n-1) * A000464(n-1).
a(n) = (2*n-1)!*[x^(2*n-1)](sec(8*x)*2*sin(4*x)). - Peter Luschny, Nov 21 2021

A349265 Generalized Euler numbers, a(n) = n![x^n](sec(5x)(sin(x) + sin(3x) + cos(2x) + cos(4x))).

Original entry on oeis.org

2, 4, 30, 272, 3522, 55744, 1066590, 23750912, 604935042, 17328937984, 551609685150, 19313964388352, 737740947722562, 30527905292468224, 1360427147514751710, 64955605537174126592, 3308161927353377294082, 179013508069217017790464, 10256718523496425979562270

Offset: 0

Peter Luschny, Nov 20 2021

nonn

For references and examples see A349264.

Matthew House, Table of n, a(n) for n = 0..377

Row 5 of A349271.

Cf. A000111, A001589, A007289, A349264, A001587.

m = 18; CoefficientList[Series[Sec[5*x] * (Sin[x] + Sin[3*x] + Cos[2*x] + Cos[4*x]), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(x) + sin(3*x) + cos(2*x) + cos(4*x))/cos(5*x)))} \\ Andrew Howroyd, Nov 20 2021

t = PowerSeriesRing(QQ, 't', default_prec=19).gen()
f = (sin(t) + sin(3*t) + cos(2*t) + cos(4*t)) / cos(5*t)
f.egf_to_ogf().list()

A349266 Generalized Euler numbers, a(n) = n![x^n](sec(7x)(-sin(2x) + sin(4x) + sin(6x) + cos(x) + cos(3x) - cos(5x))).

Original entry on oeis.org

1, 8, 64, 904, 15872, 355688, 9493504, 296327464, 10562158592, 423645846728, 18878667833344, 925434038426824, 49488442978598912, 2866986638191472168, 178867627497727197184, 11956421282992330042984, 852509723495811705208832, 64584221654333725499376008

Offset: 0

Peter Luschny, Nov 21 2021

nonn

For references and cross references, compare the overview in A349264.

Matthew House, Table of n, a(n) for n = 0..360

Row 7 of A349271.
Bisections: A064068, A064072.
Cf. A349264.

sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..17);

m = 17; CoefficientList[Series[Sec[7*x] * (-Sin[2*x] + Sin[4*x] + Sin[6*x] + Cos[x] + Cos[3*x] - Cos[5*x]), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 21 2021 *)

A349267 Generalized Euler numbers, a(n) = n![x^n](sec(8x)2(sin(4x) + cos(4x))).

Original entry on oeis.org

2, 8, 96, 1408, 29184, 739328, 22634496, 806453248, 32864600064, 1506300919808, 76717014122496, 4297849713983488, 262665886073094144, 17390688314209599488, 1239981021847665770496, 94727563504456856240128, 7719096548270543600615424, 668321603392783694711226368

Offset: 0

Peter Luschny, Nov 21 2021

nonn

For references and cross references, compare the overview in A349264.

Matthew House, Table of n, a(n) for n = 0..353

Row 8 of A349271.
Bisections: A064069, A064073.
Cf. A349264.

sec(8*x)*2*(sin(4*x) + cos(4*x)): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..17);

m = 17; CoefficientList[Series[2 * Sec[8*x] * (Sin[4*x] + Cos[4*x]), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 21 2021 *)

cp's OEIS Frontend

A000318 Generalized tangent numbers d(4,n).

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A000490 Generalized Euler numbers c(4,n).

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A001587 Generalized Euler numbers.

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A064068 Generalized Euler number c(7,n).

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A064069 Generalized Euler number c(8,n).

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A064070 Generalized Euler number c(9,n).

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Maple

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A064073 Generalized tangent number d(8,n).

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Maple

Formula

A349265 Generalized Euler numbers, a(n) = n!*[x^n](sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x))).

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A349265 Generalized Euler numbers, a(n) = n![x^n](sec(5x)(sin(x) + sin(3x) + cos(2x) + cos(4x))).

A349266 Generalized Euler numbers, a(n) = n![x^n](sec(7x)(-sin(2x) + sin(4x) + sin(6x) + cos(x) + cos(3x) - cos(5x))).

A349267 Generalized Euler numbers, a(n) = n![x^n](sec(8x)2(sin(4x) + cos(4x))).