A219613
E.g.f. tan(x/(1-x)).
Original entry on oeis.org
0, 1, 2, 8, 48, 376, 3600, 40592, 525952, 7692928, 125303040, 2248366592, 44055035904, 935800603648, 21417131939840, 525346642337792, 13748654428323840, 382362034331877376, 11260657076602208256, 350082293087247269888, 11457214800338786713600
Offset: 0
a(3) = 8: The set partitions of {1,2,3} into an odd number of blocks are {1,2,3} and {1}{2}{3}. There are 6 ways to linearly order the elements of {1,2,3}. There are 2 such ways to order the blocks of the set partition {1}{2}{3}: {1}{3}{2} and {2}{3}{1}. 6+2=8.
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nn=21;Range[0,nn]!CoefficientList[Series[Tan[x/(1-x)],{x,0,nn}],x]
A139134
a(n) = n-th derivative of Sec(Exp(z)-1) at z=0.
Original entry on oeis.org
1, 0, 1, 3, 12, 65, 417, 3094, 26243, 250371, 2652754, 30910935, 392895789, 5409783392, 80214503397, 1274329677391, 21594115259540, 388790596531173, 7411710847692157, 149142571391245758, 3159081458947418511
Offset: 0
Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008
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Table[D[Sec[Exp[x] - 1], {x, n}] /. x -> 0, {n, 0, 30}]
A317022
Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).
Original entry on oeis.org
1, 1, 2, 6, 25, 132, 838, 6209, 52592, 501238, 5308295, 61839954, 785915626, 10820482467, 160436371306, 2548722840218, 43188812459297, 777586865332600, 14823480294719570, 298285781617278681, 6318170247815155180, 140520406400556170514, 3274091838364580459623
Offset: 0
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b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(o-1+j, u-j), j=1..u))
end:
a:= n-> add(b(j, 0)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Jul 19 2018
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nmax = 22; CoefficientList[Series[Sec[Exp[x] - 1] + Tan[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
e[n_] := e[n] = (2 I)^n If[EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]; a[n_] := a[n] = Sum[StirlingS2[n, k] e[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]
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from itertools import accumulate
from sympy.functions.combinatorial.numbers import stirling
def A317022(n): # generator of terms
if n == 0: return 1
blist, c = (0,1), 0
for k in range(1,n+1):
c += stirling(n,k)*blist[-1]
blist = tuple(accumulate(reversed(blist),initial=0))
return c # Chai Wah Wu, Apr 18 2023
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]
A296543
Expansion of e.g.f. tanh(exp(x)-1).
Original entry on oeis.org
0, 1, 1, -1, -11, -33, 61, 1367, 7253, -12561, -580499, -4701497, 4669765, 580325215, 6636339165, 1365901495, -1122870368715, -17289945450289, -31110588453299, 3713822629274023, 74717183313957413, 280555705771423039, -19253195126787261507, -496715617694137066089, -3008746115751273626347
Offset: 0
tanh(exp(x)-1) = x/1! + x^2/2! - x^3/3! - 11*x^4/4! - 33*x^5/5! + 61*x^6/6! + 1367*x^7/7! + ...
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a:=series(tanh(exp(x)-1),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
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nmax = 24; CoefficientList[Series[Tanh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Sinh[Exp[x] - 1]/Cosh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[(Exp[x] - 1)/(1 + ContinuedFractionK[(Exp[x] - 1)^2, 2 k - 1, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
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