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.

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A354246 Indices of coefficients of x^(2*k-1) in Integral exp(-x*tan(x))/cos(x) dx at which the signs of the coefficients change: list of k such that sign(A354245(k)) != sign(A354245(k-1)), starting with 1.

Original entry on oeis.org

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, 1759, 1853, 1950, 2050, 2151, 2256, 2362, 2471, 2583, 2697, 2814, 2933, 3054, 3178, 3305, 3434, 3565, 3699, 3835, 3974, 4115, 4259, 4405, 4554, 4705, 4859
Offset: 1

Views

Author

Paul D. Hanna, May 20 2022

Keywords

Comments

The e.g.f. of A354245 is Integral exp(-x*tan(x))/cos(x) dx.
What is the limit of a(n)/n^2 ?
Conjecture: lim_{n->oo} a(n)/n^2 = Pi^2/8 = A111003 = 1.2337... - Vaclav Kotesovec, May 26 2022

Examples

			The expansion of Integral exp(-x*tan(x)) / cos(x) dx = 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! + ... + A354245(n)*x^(2*n-1)/(2*n-1)! + ...
The signs (+-1) of the coefficients A354245 begin:
[+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
This sequence gives the positions in A354245 at which the signs of the coefficients change.

Crossrefs

Cf. A354245, A354399, A354425.

Programs

  • Mathematica
    nmax = 500; A354245 = 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}]; Join[{1}, Select[Range[nmax], A354245[[#]]*A354245[[#-1]] < 0 &]] (* Vaclav Kotesovec, May 24 2022 *)

Extensions

a(39)-a(64) from Vaclav Kotesovec, May 26 2022

A009273 Expansion of e.g.f. exp(x*tanh(x)) (even powers only).

Original entry on oeis.org

1, 2, 4, -24, 400, -5600, -103872, 26975872, -3438685952, 417995260416, -51382607559680, 5994623640856576, -454930757753597952, -94991612229069430784, 81515752167646959124480, -41079088828539119883878400, 18870487103065970636941754368, -8553231336572387307575081566208
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Crossrefs

Cf. A354245, A354399.

Programs

  • Mathematica
    nmax = 20; Table[(CoefficientList[Series[E^(x*Tanh[x]), {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}] (* Vaclav Kotesovec, May 24 2022 *)
With[{nn=40},Take[CoefficientList[Series[Exp[x Tanh[x]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Apr 02 2025 *)
  • Maxima
    a(n):=sum(binomial(2*n,m)*sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(2*n-2*m-k)*stirling2(2*n-m,k+m),k,0,2*n-2*m),m,0,2*n); /* Vladimir Kruchinin, Jun 06 2011 */

Formula

a(n) = Sum_(m=0..2*n, binomial(2*n,m)*Sum_(k=0..2*n-2*m, binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(2*n-2*m-k)*stirling2(2*n-m,k+m))), n>0, a(0)=1. - Vladimir Kruchinin, Jun 06 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
More terms from Vaclav Kotesovec, May 25 2022

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 1-3 of 3 results.

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