A003710
Expansion of e.g.f. cos(tan(x)) (even powers only).
Original entry on oeis.org
1, -1, -7, -97, -2063, -53409, -752343, 166831871, 43685848289, 9398558916159, 2116926930779225, 524586454143030495, 144620290378876829905, 44287070229737735633567, 14954349885478653319004041
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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nn = 20; Table[(CoefficientList[Series[Cos[Tan[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)
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a(n):=if n=0 then 1 else 2*sum((sum(binomial(j-1,2*m-1)*j!*2^(2*n-j-1)*(-1)^(n+j)*stirling2(2*n,j),j,2*m,2*n))/(2*m)!,m,0,n); /* Vladimir Kruchinin, Jun 28 2011 */
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my(x='x+O('x^30)); select(x->x, Vec(serlaplace(cos(tan(x))))) \\ Michel Marcus, Oct 02 2021
A081443
Binomial transform of expansion of cosh(sinh(x)).
Original entry on oeis.org
1, 1, 2, 4, 12, 36, 128, 456, 1872, 7888, 37600, 184064, 990784, 5444544, 32333824, 195982208, 1272660224, 8441139456, 59527313920, 428299217920, 3252626013184, 25165446157312, 204354574172160, 1689266143553536, 14594815769038848
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x)*Cosh(Sinh(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 14 2019
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seq(coeff(series(exp(x)*cosh(sinh(x)), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 14 2019
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With[{nn=30},CoefficientList[Series[Exp[x]Cosh[Sinh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 14 2011 *)
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my(x='x+O('x^30)); Vec(serlaplace( exp(x)*cosh(sinh(x)) )) \\ G. C. Greubel, Aug 14 2019
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[factorial(n)*( exp(x)*cosh(sinh(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 14 2019
A081444
Second binomial transform of expansion of cosh(sinh(x)).
Original entry on oeis.org
1, 2, 5, 14, 45, 162, 641, 2718, 12249, 58370, 294493, 1571374, 8858053, 52542882, 326804185, 2122860862, 14366452913, 101063123202, 738075208501, 5587492946510, 43795838894941, 354876969914786, 2968890716640945
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(2*x)*Cosh(Sinh(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 14 2019
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seq(coeff(series(exp(2*x)*cosh(sinh(x)), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 14 2019
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With[{nn = 30}, CoefficientList[Series[Exp[2x]Cosh[Sinh[x]], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
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my(x='x+O('x^30)); Vec(serlaplace( exp(2*x)*cosh(sinh(x)) )) \\ G. C. Greubel, Aug 14 2019
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[factorial(n)*( exp(2*x)*cosh(sinh(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 14 2019
A297214
Expansion of e.g.f. exp(cos(sin(x))-1) (even powers only).
Original entry on oeis.org
1, -1, 8, -127, 3523, -146964, 8538477, -655457233, 63974756924, -7713566822979, 1123255462229507, -193995005614903728, 39147722262966666217, -9115873617718182207793, 2423565558533387761866928, -728969374928760685473620951, 246100624914698937364249220851
Offset: 0
exp(cos(sin(x))-1) = 1 - x^2/2! + 8*x^4/4! - 127*x^6/6! + 3523*x^8/8! - 146964*x^10/10! + ...
Cf.
A003709,
A009045,
A009201,
A009202,
A009203,
A009204,
A009238,
A009239,
A009240,
A009241,
A009254,
A297215.
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nmax = 16; Table[(CoefficientList[Series[Exp[Cos[Sin[x]] - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
Showing 1-4 of 4 results.
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