A061697
Generalized Bell numbers.
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 201, 1226, 5587, 493333, 8910253, 109739620, 6832444928, 251336859489, 6402632091649, 369288128260091, 21333939590516867, 941843896620169405, 60266201588496408645, 4623833509894543300868, 309412778502377193367456, 24102475277979402591991181
Offset: 0
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}]
A346271
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^2 / 4 ).
Original entry on oeis.org
1, 1, 2, 7, 41, 346, 3807, 53747, 952275, 20362552, 515112983, 15277888693, 523304644304, 20415373547609, 900219731675981, 44533809102813206, 2451041479421900803, 149140880201760643360, 9982798939295116151967, 731215136812226462200109, 58333374310397488522052976
Offset: 0
-
nmax = 20; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 20}]
A346272
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^3 / 36 ).
Original entry on oeis.org
1, 1, 3, 15, 115, 1196, 16282, 276158, 5713507, 140482000, 4047179258, 134447125418, 5097852537802, 218254682152053, 10469861372693621, 558373926672949031, 32908746221003292003, 2130712239317226923136, 150826951188229240683858, 11618459541824256750732794
Offset: 0
-
nmax = 19; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^3/36], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = a[1] = 1; a[n_] := a[n] = n a[n - 1] + n (n - 1)^2 a[n - 2]/2 + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 19}]
Showing 1-4 of 4 results.