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
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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 *)
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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 */
A354425
List of k such that sign(A009277(k)) = sign(A009277(k+1)).
Original entry on oeis.org
0, 2, 6, 10, 16, 22, 29, 37, 45, 54, 63, 73, 83, 93, 104, 116, 128, 140, 153, 166, 179, 193, 207, 221, 236, 251, 266, 282, 298, 314, 331, 347, 364, 382, 399, 417, 435, 454, 473, 491, 511, 530, 550, 570, 590, 610, 631, 652, 673, 694, 715, 737, 759, 781, 804, 826, 849, 872, 895, 919, 942, 966, 990
Offset: 1
2 is in the sequence because A009277(2) = -4 and A009277(3) = -88 have the same sign.
6 is in the sequence because A009277(6) = 675776 and A009277(7) = 903834752 have the same sign.
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nmax = 500; A009277 = Table[(CoefficientList[Series[Exp[Tanh[x]^2], {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[k]], {k, 3, 2*nmax, 2}]; Join[{0}, Select[Range[nmax-2], A009277[[#]]*A009277[[#+1]] > 0 &]]
With[{nn=2000},SequencePosition[Sign[Take[CoefficientList[Series[Exp[Tanh[x]^2],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]],{x_,x_}]][[;;,1]]-1 (* Harvey P. Dale, Apr 08 2023 *)
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
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.
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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 *)
Showing 1-3 of 3 results.
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