A064068 -id:A064068 - 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

A064072 Generalized tangent number d(7,n).

Original entry on oeis.org

8, 904, 355688, 296327464, 423645846728, 925434038426824, 2866986638191472168, 11956421282992330042984, 64584221654333725499376008, 438640634423372575622395751944, 3658596185733807024739320857622248, 36763878323837308563984663576886049704

Offset: 1

Eric W. Weisstein, Aug 31 2001

Lars Blomberg, Table of n, a(n) for n = 1..180
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 1967 663-688.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699.
Eric Weisstein's World of Mathematics, Tangent Number.

Cf. A064068, A349266, A349264.

egf := sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x)): ser := series(egf, x, 24):
seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..10); # Peter Luschny, Nov 21 2021

a(n) = (2*n-1)!*[x^(2*n-1)](sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x))). - Peter Luschny, Nov 21 2021

Offset changed to 1 by Lars Blomberg, Sep 07 2015

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 *)

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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A064072 Generalized tangent number d(7,n).

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

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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)).

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