A349265 -id:A349265 - 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

A000187 Generalized Euler numbers, c(5,n).

Original entry on oeis.org

2, 30, 3522, 1066590, 604935042, 551609685150, 737740947722562, 1360427147514751710, 3308161927353377294082, 10256718523496425979562270, 39490468691102039103925777602, 184856411587530526077816051412830, 1033888847501229495999134528615701122

Offset: 0

N. J. A. Sloane

			a(3) = 1066590: L_5(7) = Sum_{n >= 0} (-1)^n*( 1/(10*n+1)^7 + 1/(10*n+3)^7 + 1/(10*n+7)^7 + 1/(10*n+9)^7 ) = 1066590*( (1/6!)*sqrt(5)*(Pi/10)^7 ). - _Peter Bala_, Nov 18 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).

Sean A. Irvine, Table of n, a(n) for n = 0..250
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 5 of A235605.

Cf. A000192, A000490, A060096, A000320, A349265, A349264.

seq((-1)^n*(10)^(2*n)*(euler(2*n,1/10) + euler(2*n,3/10)), n = 0..11); # Peter Bala, Nov 18 2020
egf := sec(5*x)*(cos(2*x) + cos(4*x)): ser := series(egf, x, 26):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # Peter Luschny, Nov 21 2021

a0=5; nmax=20; km0 = nmax; Clear[cc]; L[a_, s_, km_] := Sum[JacobiSymbol[ -a, 2k+1]/(2k+1)^s, {k, 0, km}]; c[a_, n_, km_] := 2^(2n+1)*Pi^(-2n-1)*(2n)!*a^(2n+1/2)*L[a, 2n+1, km] // Round; cc[km_] := cc[km] = Table[ c[a0, n, km], {n, 0, nmax}]; cc[km0]; cc[km = 2km0]; While[cc[km] != cc[ km/2, km = 2km]]; A000187 = cc[km] (* Jean-François Alcover, Feb 05 2016 *)

From the Shanks paper: 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 c_(a,n) are defined by L_a(2n+1)= (Pi/(2a))^(2n+1)*sqrt(a)* c(a,n)/ (2n)! for a > 1 and n = 0,1,2,... - Sean A. Irvine, Mar 26 2012
From Peter Bala, Nov 18 2020: (Start)
a(n) = (-1)^n*10^(2*n)*( E(2*n,1/10) + E(2*n,3/10) ), where E(n,x) are the Euler polynomials - see A060096.
Row 5 of A235605.
G.f.: A(x) = 2*cos(x)*cos(3*x)/( 2*cos(x)*cos(4*x) - cos(3*x) ) = 2 + 30*x^2/2! + 3522*x^4/4! + ....
Alternative forms:
A(x) = (exp(i*x) + exp(3*i*x) + exp(7*i*x) + exp(9*i*x))/(1 + exp(10*i*x));
A(x) = (sqrt(5)/10)*( sec(x + Pi/5) + sec(x + 2*Pi/5) - sec(x + 3*Pi/5) - sec(x + 4*Pi/5) ).  (End)
a(n) = (2*n)!*[x^(2*n)](sec(5*x)*(cos(2*x) + cos(4*x))). - Peter Luschny, Nov 21 2021
a(n) ~ 2^(4*n + 2) * 5^(2*n + 1/2) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 15 2022

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

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).

A000320 Generalized tangent numbers d(5,n).

Original entry on oeis.org

4, 272, 55744, 23750912, 17328937984, 19313964388352, 30527905292468224, 64955605537174126592, 179013508069217017790464, 620314831396713435870789632, 2639743384489464189324523208704, 13533573366345611477262311433961472, 82274260343572247169162187576069586944

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..189
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. A000318, A000187, A349265, A349264.

egf := sec(5*x)*(sin(x) + sin(3*x)): ser := series(egf, x, 26):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..13); # Peter Luschny, Nov 21 2021

nmax = 15; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k + 1]/(2k+1)^s, {k, 0, km}]; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[5, n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000320 = dd[km] (* Jean-François Alcover, Feb 07 2016 *)

a(n) = (2*n-1)!*[x^(2*n-1)](sec(5*x)*(sin(x) + sin(3*x))). - Peter Luschny, Nov 21 2021

Formula producing A000326, rather than this sequence, deleted by Sean A. Irvine, Sep 09 2010

a(10)-a(13) from Lars Blomberg, Sep 07 2015

cp's OEIS Frontend

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

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A000187 Generalized Euler numbers, c(5,n).

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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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A000320 Generalized tangent numbers d(5,n).

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