A365010
E.g.f. satisfies A(x) = 1 + x*exp(-x)*A(x)^3.
Original entry on oeis.org
1, 1, 4, 39, 596, 12365, 324714, 10329655, 386190328, 16597810233, 806356830230, 43700423019011, 2613919719004692, 171053575111641157, 12156558707970920866, 932424974682447304815, 76772968644326739801584, 6754080601542663692950769
Offset: 0
-
A365010 := proc(n)
add( (-k)^(n-k)*A001764(k)/(n-k)!,k=0..n) ;
%*n! ;
end proc:
seq(A365010(n),n=0..80); # R. J. Mathar, Aug 16 2023
-
a(n) = n!*sum(k=0, n, (-k)^(n-k)*binomial(3*k, k)/((2*k+1)*(n-k)!));
A368272
Expansion of e.g.f. exp(-x) / (1 + x*exp(x)).
Original entry on oeis.org
1, -2, 3, -1, -11, 19, 151, -799, -2295, 37367, -16469, -2114531, 9695533, 132142451, -1556927553, -6822608311, 234527654161, -360436983569, -35798255259821, 294290464165685, 5217729367883061, -102317187098688661, -517822188623299097, 31412148276241662049
Offset: 0
A368273
Expansion of e.g.f. exp(-2*x) / (1 + x*exp(x)).
Original entry on oeis.org
1, -3, 8, -17, 20, 23, -50, -1185, 6648, 20143, -372646, 179111, 25378468, -126050121, -1849977930, 23353880527, 109161798512, -3986970251809, 6487865966386, 680166849412311, -5885809282265124, -109572316727641433, 2250978116175344846, 11909910338327490623
Offset: 0
A335577
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
Original entry on oeis.org
1, -1, -2, 9, 32, -285, -1236, 18725, 86176, -2087001, -9204580, 351964569, 1336442304, -83422970917, -231889447076, 26389118293005, 35917342192064, -10722110983670193, 5028963509133756, 5432569724760331841, -14852185163192897120, -3352369390318855889661
Offset: 0
-
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
nmax = 21; CoefficientList[Series[1/(1 + Exp[x] x (1 + x)), {x, 0, nmax}], x] Range[0, nmax]!
A335578
a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).
Original entry on oeis.org
1, -1, -6, 15, 272, -745, -29976, 61271, 6065856, -2723697, -1941455080, -3989345041, 897021218400, 4964061925511, -562221881675832, -5689641396555705, 456732442022509184, 7321841133968133023, -464200472167634521800, -10961686347887871324289, 573373115861405030522400
Offset: 0
-
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[1/(1 + Exp[x] x (1 + 3 x + x^2)), {x, 0, nmax}], x] Range[0, nmax]!
A365011
E.g.f. satisfies A(x) = 1 + x*exp(-x)*A(x)^4.
Original entry on oeis.org
1, 1, 6, 87, 1964, 60325, 2349114, 110922091, 6159510552, 393373489257, 28407518470070, 2289019332293551, 203608076603605860, 19816972252710998989, 2094926215725519979698, 239037380421621120397395, 29281119335188021375533104, 3832665229749097186190010193
Offset: 0
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
-
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}]