A000490 -id:A000490 - 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

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

Original entry on oeis.org

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

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

A000318 Generalized tangent numbers d(4,n).

Original entry on oeis.org

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

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

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

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A000318 Generalized tangent numbers d(4,n).

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