cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A354399 List of k such that sign(A009273(k)) = sign(A009273(k+1)).

Original entry on oeis.org

0, 1, 5, 12, 21, 33, 47, 64, 83, 105, 129, 155, 184, 216, 250, 286, 325, 366, 410, 456, 505, 556, 610, 666, 725, 786, 849, 915, 984, 1055, 1128, 1204, 1282, 1363, 1446, 1532, 1620, 1711, 1804, 1900, 1998, 2098, 2201, 2307, 2415, 2525, 2638, 2753, 2871, 2991, 3114, 3239, 3367, 3497, 3630, 3765, 3903, 4043, 4185, 4330, 4477, 4627, 4780, 4935
Offset: 1

Views

Author

Vaclav Kotesovec, May 25 2022

Keywords

Comments

Conjecture: lim_{n->oo} a(n)/n^2 = Pi^2/8 = A111003 = 1.2337...

Examples

			12 is in the sequence because A009273(12) = -454930757753597952 and A009273(13) = -94991612229069430784 have the same sign.

Crossrefs

Cf. A009273, A354246, A354425.

Programs

  • Mathematica
    nmax = 400; A009273 = Table[(CoefficientList[Series[E^(x*Tanh[x]), {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}]; Join[{0}, Select[Range[nmax-1], A009273[[#]]*A009273[[#-1]] > 0 &] - 1]

A354245 E.g.f.: Integral exp(-x*tan(x)) / cos(x) dx.

Original entry on oeis.org

1, -1, -3, -5, 441, 25911, 1384757, 74436531, 3175224945, -135369432209, -89771310955155, -25527579751884693, -6567045994040209879, -1678101422880410465625, -427686430807976068014939, -102728760825086263958009309, -18156608776369804213731821343, 2585946334251026101959272934111
Offset: 1

Views

Author

Paul D. Hanna, May 20 2022

Keywords

Comments

The positions at which the signs of the terms change appear to be ~ c*n^2 (see example section).

Examples

			E.g.f.: A(x) = x - x^3/3! - 3*x^5/5! - 5*x^7/7! + 441*x^9/9! + 25911*x^11/11! + 1384757*x^13/13! + 74436531*x^15/15! + 3175224945*x^17/17! - 135369432209*x^19/19! + ...
where d/dx A(x) = exp(-x*tan(x)) / cos(x).
Also, e.g.f. A(x) equals the limit of the finite sum:
A(x) = lim_{N->oo} (x/N) * [1 + cos(2*x/N)/cos(x/N)^2 + cos(3*x/N)^2/cos(2*x/N)^3 + cos(4*x/N)^3/cos(3*x/N)^4 + cos(5*x/N)^4/cos(4*x/N)^5 + cos(6*x/N)^5/cos(5*x/N)^6 + ... + cos(x)^(N-1)/cos((N-1)*x/N)^N].
PATTERN OF SIGNS.
The signs (+-1) of the terms begin:
[+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
The positions at which the signs of the terms change begin as follows:
[1, 2, 5, 10, 18, 29, 42, 57, 75, 95, 118, 143, 171, 201, 234, 269, 307, 347, 390, 435, 482, 532, 585, 639, 697, 757, 819, 884, 951, 1021, 1093, 1167, 1245, 1324, 1406, 1491, 1578, 1667, ..., A354246(n), ...]
which appears to be asymptotic to c*n^2 for some constant c ~ 1.2...

Links

Crossrefs

Cf. A009244, A009264, A009273, A354020, A354246.

Programs

  • Mathematica
    nmax = 20; Table[(CoefficientList[Series[1/(E^(x*Tan[x])*Cos[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}] (* Vaclav Kotesovec, May 24 2022 *)
  • PARI
    {a(n) = my(A = intformal( exp(-x * tan(x +O(x^(2*n+1))))/cos(x +O(x^(2*n+1)) ) )); (2*n-1)!*polcoeff(A,2*n-1)}
for(n=1,20,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! may be defined by:
(1) A(x) = Integral exp(-x*tan(x)) / cos(x) dx.
(2) A(x) = lim_{N->oo} (x/N) * Sum_{n=1..N} cos((n+1)*x/N)^n / cos(n*x/N)^(n+1).

A296835 Expansion of e.g.f. exp(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 1, 4, 33, 451, 9110, 253401, 9246881, 427272364, 24332740569, 1671761966755, 136185663849422, 12966840876896193, 1425738305622057713, 179172604156015950676, 25507107918052543195905, 4081610970381242583997171, 729135575105289450378655526
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			exp(x*tan(x/2)) = 1 + x^2/2! + 4*x^4/4! + 33*x^6/6! + 451*x^8/8! + ...

Crossrefs

Cf. A001469, A003723, A006229, A009252, A009273, A110501, A296836.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[Exp[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*tan(x/2)).

A296787 Expansion of e.g.f. exp(x*arctan(x)) (even powers only).

Original entry on oeis.org

1, 2, 4, 24, -496, 36000, -3753408, 556961664, -111591202560, 29054584410624, -9541382573767680, 3858875286730168320, -1884995591107521540096, 1094305223336273239449600, -744771228363250138965196800, 587358379156469629707528929280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			exp(x*arctan(x)) = 1 + 2*x^2/2! + 4*x^4/4! + 24*x^6/6! - 496*x^8/8! + ...

Crossrefs

Cf. A002019, A009252, A009273, A010050, A102059, A166356, A259647, A293192, A296788, A296789.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTan[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[(I/2) x (Log[1 - I x] - Log[1 + I x])], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*arctan(x)).
a(n) ~ -(-1)^n * 2^(2*n-1) * n^(2*n-1) / exp(2*n). - Vaclav Kotesovec, Dec 21 2017

A296789 Expansion of e.g.f. exp(x*arctanh(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 504, 24464, 1959840, 234852672, 39370660224, 8799246209280, 2528787321598464, 908585701684024320, 399070678264750356480, 210373049449102957645824, 131083661069772517440921600, 95304505860052894815543705600, 79961055068441273887848131297280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			exp(x*arctanh(x)) = 1 + 2*x^2/2! + 20*x^4/4! + 504*x^6/6! + 24464*x^8/8! + ...

Crossrefs

Cf. A000246, A009252, A009273, A010050, A166356, A259647, A293193, A296787, A296788.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[x (Log[1 + x] - Log[1 - x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*arctanh(x)).
a(n) ~ 2^(2*n + 2) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Dec 21 2017

A296836 Expansion of e.g.f. exp(x*tanh(x/2)) (even powers only).

Original entry on oeis.org

1, 1, 2, 3, -3, 20, 105, -5271, 133826, -2714517, 25525845, 2131781300, -235250824479, 17527695547713, -1124258412169438, 58383380825728035, -975024061456732035, -398903577787777972396, 97649546210035758250281, -17069419358223320552890167
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			exp(x*tanh(x/2)) = 1 + x^2/2! + 2*x^4/4! + 3*x^6/6! - 3*x^8/8! + ...

Crossrefs

Cf. A001469, A003723, A006229, A009252, A009273, A110501, A296835.

Programs

  • Mathematica
    nmax = 19; Table[(CoefficientList[Series[Exp[x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*tanh(x/2)).

A355157 E.g.f.: exp(-x*tan(x)/2).

Original entry on oeis.org

1, -1, -1, -3, 81, 5815, 367791, 26531589, 2231306849, 213451322031, 21883750567455, 2048495154903373, 38238322254619953, -84508521173113956633, -48971496478118456699569, -23204642085706390327411755, -10849535606183313563987736639, -5259600981361065001871911307681, -2687430634156750207499977165012161
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2022

Keywords

Examples

			E.g.f. A(x) = 1 - x^2/2! - x^4/4! - 3*x^6/6! + 81*x^8/8! + 5815*x^10/10! + 367791*x^12/12! + 26531589*x^14/14! + 2231306849*x^16/16! + 213451322031*x^18/18! + 21883750567455*x^20/20! + 2048495154903373*x^22/22! + 38238322254619953*x^24/24! - 84508521173113956633*x^26/26! + ...
The signs (+/-) of the terms a(n) begin:
[+,
-,-,-,
+,+,+,+,+,+,+,+,+,
-,-,-,-,-,-,-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,
-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,
-, ...].
The change in signs take place at positions:
[1, 4, 13, 26, 45, 68, 97, 130, 168, 211, 259, 312, 370, 433, 501, 574, 652, 734, 822, 914, 1012, 1114, 1222, 1334, 1451, 1573, 1700, 1832, 1969, 2111, 2258, 2410, 2566, 2728, 2895, ...].

Crossrefs

Cf. A354245, A009273.

Programs

  • PARI
    {a(n) = my(A = exp(-x * tan(x +O(x^(2*n+1)))/2) ); (2*n)!*polcoeff(A,2*n)}
for(n=0,20,print1(a(n),", "))
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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