A336210
a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k).
Original entry on oeis.org
1, -1, 3, -10, -117, 5224, -23010, -10319891, 463834315, 69461529092, -10005601418172, -1323175060249241, 468450359815048182, 63281374513705043227, -46495538420749056681263, -7147072328212024308730535, 9119277358213513566069911755, 1827085356172328516064256064092
Offset: 0
-
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^3 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[-Sum[x^k/(k!)^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3
A337166
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 + x - BesselI(0,2*sqrt(x))).
Original entry on oeis.org
1, 0, -1, -1, 17, 99, -926, -20385, 25969, 7206059, 90298826, -3271747557, -149187119280, 236884125841, 233237751740057, 7110791842650002, -293292401726383791, -32980038867059802549, -498084376275585698222, 114298048468067933019627, 9072219653673352772098960
Offset: 0
-
A337166 := proc(n)
option remember ;
if n = 0 then
1;
else
add(binomial(n,k)^2*(n-k)*procname(k),k=0..n-2) ;
-%/n ;
end if;
simplify(%) ;
end proc:
seq(A337166(n),n=0..40) ; # R. J. Mathar, Aug 19 2022
-
nmax = 20; CoefficientList[Series[Exp[1 + x - BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 20}]
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}]
A336831
a(n) = (n!)^n * [x^n] exp(-Sum_{k>=1} (-x)^k / (k!)^n).
Original entry on oeis.org
1, 1, 1, 10, 4359, 91406876, 111657668637280, 11436881770074767723291, 137560155520600195617494951186559, 260122627893213770028102613184254361777327032, 99781796293430843492956500115058179262448159117567276656136
Offset: 0
-
Table[(n!)^n SeriesCoefficient[Exp[-Sum[(-x)^k/(k!)^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := b[n, k] = If[n == 0, 1, -Sum[(-1)^(n - j) Binomial[n, j]^k (n - j) b[j, k], {j, 0, n - 1}]/n]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]
Showing 1-4 of 4 results.