A349267 -id:A349267 - cp's OEIS frontend

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

1, 4, 16, 128, 1280, 16384, 249856, 4456448, 90767360, 2080374784, 52975108096, 1483911200768, 45344872202240, 1501108249821184, 53515555843342336, 2044143848640217088, 83285910482761809920, 3605459138582973251584, 165262072909347030040576, 7995891855149741436305408

Offset: 0

Peter Luschny, Nov 20 2021

nonn

			Exponential generating functions of generalized Euler numbers in context:
egf1 = sec(1*x)*(sin(x) + 1).
   [A000111, A000364, A000182]
egf2 = sec(2*x)*(sin(x) + cos(x)).
   [A001586, A000281, A000464]
egf3 = sec(3*x)*(sin(2*x) + cos(x)).
   [A007289, A000436, A000191]
egf4 = sec(4*x)*(sin(4*x) + 1).
   [A349264, A000490, A000318]
egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).
   [A349265, A000187, A000320]
egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).
   [A001587, A000192, A000411]
egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).
   [A349266, A064068, A064072]
egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).
   [A349267, A064069, A064073]
egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.
   [A349268, A064070, A064074]

Matthew House, Table of n, a(n) for n = 0..389
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, 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]

Row 4 of A349271.
Cf. A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
Secant  bisections: A000364, A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070.
Tangent bisections: A000182, A000464, A000191, A000318, A000320, A000411, A064072, A064073, A064074.
Cf. A225147, A349429.

sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);

m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

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

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).

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

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

cp's OEIS Frontend

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

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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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A064069 Generalized Euler number c(8,n).

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A064073 Generalized tangent number d(8,n).

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cp's OEIS Frontend

A349264 Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).

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

Views

Author

Keywords

Examples

Links

Crossrefs

A064069 Generalized Euler number c(8,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A064073 Generalized tangent number d(8,n).

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Author

Keywords

Links

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Programs

Maple

Formula

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).