A337591
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^3 * a(n-k).
Original entry on oeis.org
1, 1, 6, 51, 760, 15545, 428256, 15043483, 653049664, 34204348305, 2118834917200, 152834879685851, 12670536337934256, 1194143629239156505, 126753440317516749660, 15031687739886065433375, 1977667235694725269563136, 286890421090357737699794209, 45637300134026406622214264592
Offset: 0
-
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
A337590
a(0) = 0; a(n) = n - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k) * k * a(k).
Original entry on oeis.org
0, 1, 0, -3, 28, -215, -174, 90223, -3840472, 103719537, 429704110, -357346077869, 35100093531900, -2005608652057595, -24108041118593418, 27881407632242902515, -4876442148527153942384, 474102062424164433715937, 12637408141631813073125094, -18867461801192524662360616421
Offset: 0
-
a[0] = 0; a[n_] := a[n] = n - (1/n) Sum[Binomial[n, k]^2 (n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Log[1 + Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
A337826
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).
Original entry on oeis.org
1, 1, 10, 105, 2248, 62445, 2390436, 116650177, 7043659904, 514744959321, 44534754680500, 4493090921151261, 521600149636044480, 68900819660071184149, 10259571068808850618480, 1708054303772376318547125, 315688007001129064574027776, 64370788231256983836207599153
Offset: 0
-
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k^4 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2
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}]
A352658
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * binomial(k+1,2) * k * a(n-k).
Original entry on oeis.org
1, 1, 5, 39, 508, 9235, 224481, 6959932, 266492388, 12302514945, 671505310855, 42664357009186, 3114726872133570, 258452373177094213, 24149855477595375815, 2520813303733886387220, 291892618561012451083816, 37264133443594227118861233, 5216461719269145457350349359
Offset: 0
-
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[k + 1, 2] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(x BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
Showing 1-5 of 5 results.