A356827
Expansion of e.g.f. exp(x * exp(3*x)).
Original entry on oeis.org
1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722
Offset: 0
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A356827 := proc(n)
add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
end proc:
seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))
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a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));
A357948
Expansion of e.g.f. exp( x * exp(-x^2) ).
Original entry on oeis.org
1, 1, 1, -5, -23, 1, 601, 2731, -13775, -219743, -313199, 15383611, 125451481, -811558175, -20767068503, -37852036949, 2898343066081, 28990920216001, -313289894357855, -8634009894555653, -3214642669500599, 2108734127922999361, 20183394611962437241
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-x^2))))
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a(n) = n!*sum(k=0, n\2, (-n+2*k)^k/(k!*(n-2*k)!));
A316159
Expansion of e.g.f. exp(exp(x*exp(-x)) - 1).
Original entry on oeis.org
1, 1, 0, -4, -1, 47, 17, -1111, -12, 43476, -49665, -2391805, 7528897, 168436465, -1052303380, -14234148280, 161462347715, 1288890088835, -27585406164839, -91839429007223, 5125915000647712, -6443738757309888, -1013794188308572677, 6728499674632962055, 205866724424357904465
Offset: 0
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
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a:=series(exp(exp(x*exp(-x))-1),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[Exp[Exp[x Exp[-x]] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[(-k)^(n - k) Binomial[n, k] BellB[k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]
A328488
Expansion of e.g.f. 1 / (2 - exp(x * exp(x))).
Original entry on oeis.org
1, 1, 5, 34, 307, 3456, 46659, 734882, 13227995, 267871036, 6027206803, 149176155030, 4027831914099, 117816299188472, 3711283196035523, 125258162280991858, 4509378597919760779, 172486973301491042964, 6985853719202139488211, 298650859698906574479278
Offset: 0
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nmax = 19; CoefficientList[Series[1/(2 - Exp[x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
A336610
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-sqrt(x) * BesselI(1,2*sqrt(x))).
Original entry on oeis.org
1, -1, 0, 9, -4, -625, -906, 145187, 1350040, -71822385, -2093778910, 49843036199, 4422338360340, 7491520000835, -11939082153832302, -455740256735697165, 33146485198521406064, 4039886119274766333343, 2019781328116371668154
Offset: 0
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nmax = 18; CoefficientList[Series[Exp[-Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -n Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]