A080832
Expansion of e.g.f. exp(x) * (sec(exp(x) - 1))^2.
Original entry on oeis.org
1, 1, 3, 13, 67, 421, 3115, 26349, 250867, 2655541, 30929019, 393019837, 5410699075, 80221867909, 1274393162827, 21594697199757, 388796268801427, 7411769447027413, 149143210226032923, 3159088788867736669
Offset: 0
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seq(coeff(series(factorial(n)*exp(x)*(sec(exp(x)-1))^2, x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 28 2018
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nn=21;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[Tan[Exp[x]-1],{x,0,nn}],x],1] (* Geoffrey Critzer, Nov 23 2012 *)
A317409
Expansion of e.g.f. cos(x/(1 - x)).
Original entry on oeis.org
1, 0, -1, -6, -35, -220, -1501, -10962, -83495, -632952, -4260601, -13852190, 355180981, 12991115436, 320077652075, 7153866992790, 155785273182001, 3395838000334352, 75000970329466895, 1687941779356532682, 38803334491247820301, 911633573138881234740, 21870615120012355726259
Offset: 0
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a:=series(cos(x/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Cos[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k Binomial[n - 1, 2 k - 1] n!/(2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
Join[{1}, Table[(1 - n) n! HypergeometricPFQ[{1 - n/2, 3/2 - n/2}, {3/2, 3/2, 2}, -1/4]/2, {n, 22}]]
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my(x='x + O('x^25)); Vec(serlaplace(cos(x/(1 - x)))) \\ Michel Marcus, Mar 26 2019
A228841
E.g.f.: sec(sec(x)-1) (even-indexed coefficients only).
Original entry on oeis.org
1, 0, 3, 75, 3108, 205125, 19839633, 2643131400, 463873573803, 103710628476075, 28775903316814668, 9702563010998171325, 3907429085273025561153, 1852516229654506870381200, 1021325008815288529961197683, 647900078249178232882473232875
Offset: 0
a(3) = 75. There are 15 set partitions of {1,2,3,4,5,6} that have an even number of even sized blocks Cf. A059386. They all have the same structure: 2/4. We build a zig permutation on each block in 1*5=5 ways. For each of these we then build a zig permutation on a representative from each of the 2 blocks in 1 way. So 5*1=5 and there are 15 such partitions so 5 *15 =75.
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nn=30;Insert[Select[Range[0,nn]!CoefficientList[Series[Sec[Sec[x]-1],{x,0,nn}],x],#>0&],0,2]
A317406
Expansion of e.g.f. sin(x/(1 - x)).
Original entry on oeis.org
0, 1, 2, 5, 12, 1, -450, -6931, -89096, -1120895, -14394150, -191263051, -2638282812, -37716883775, -556075744042, -8385570334051, -127637336779920, -1916072623603199, -27033275598036174, -311878728377256475, -918069644450841860, 120594465496571606401, 6362190374664242284782
Offset: 0
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a:=series(sin(x/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Sin[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(k + 1) Binomial[n - 1, 2 k - 2] n!/(2 k - 1)!, {k, Floor[(n + 1)/2]}], {n, 0, 22}]
Join[{0}, Table[n! HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {1/2, 1, 3/2}, -1/4], {n, 22}]]
A317411
Expansion of e.g.f. tanh(x/(1 - x)).
Original entry on oeis.org
0, 1, 2, 4, 0, -104, -1200, -10352, -68992, -177536, 5310720, 145374208, 2512029696, 33484862464, 294806779904, -1053894275072, -133585065738240, -4148745576218624, -92116699894185984, -1510783598900412416, -11327731683300474880, 414155615537321476096, 25541948643911385219072
Offset: 0
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a:=series(tanh(x/(1 - x)), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Tanh[x/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] 2^k EulerE[k, 1] n!/k!, {k, n}], {n, 0, 22}]
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