A294312
Expansion of e.g.f. sec(x*exp(x)).
Original entry on oeis.org
1, 0, 1, 6, 29, 180, 1501, 14434, 154265, 1856232, 24953401, 368767102, 5936244533, 103519338780, 1944554725205, 39134556793050, 840024295910833, 19157944025344464, 462629389438242673, 11792248121970820598, 316398168231432879565, 8913743651504295251844
Offset: 0
sec(x*exp(x)) = 1 + x^2/2! + 6*x^3/3! + 29*x^4/4! + 180*x^5/5! + 1501*x^6/6! + ...
Cf.
A000364,
A009007,
A009017,
A009121,
A009300,
A009448,
A009565,
A009635,
A009768,
A139134,
A191719,
A216401,
A217502,
A294313,
A297009,
A297010.
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a:=series(sec(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
-
nmax = 21; CoefficientList[Series[Sec[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cos[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
A306336
Expansion of e.g.f. sec(log(1 + x)) + tan(log(1 + x)).
Original entry on oeis.org
1, 1, 0, 1, -2, 10, -50, 320, -2340, 19640, -184900, 1932500, -22187200, 277576000, -3757884000, 54732418000, -853278998000, 14176686784000, -250046057846000, 4665989766386000, -91838330641200000, 1901405069222360000, -41307212202493120000, 939523370329035440000, -22327292561388519640000
Offset: 0
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a:=series(sec(log(1 + x)) + tan(log(1 + x)),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 26 2019
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nmax = 24; CoefficientList[Series[Sec[Log[1 + x]] + Tan[Log[1 + x]], {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[StirlingS1[n, k] e[k], {k, 0, n}]; Table[a[n], {n, 0, 24}]
-
from itertools import accumulate
from sympy.functions.combinatorial.numbers import stirling
def A306336(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,kind=1,signed=True)*blist[-1]
blist = tuple(accumulate(reversed(blist),initial=0))
return c # Chai Wah Wu, Apr 18 2023
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