A337166 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 + x - BesselI(0,2*sqrt(x))).

1, 0, -1, -1, 17, 99, -926, -20385, 25969, 7206059, 90298826, -3271747557, -149187119280, 236884125841, 233237751740057, 7110791842650002, -293292401726383791, -32980038867059802549, -498084376275585698222, 114298048468067933019627, 9072219653673352772098960

Offset: 0

Author

Ilya Gutkovskiy, Jan 28 2021

Keywords

• sign

Crossrefs

Cf. A061696, A293037, A336209.

Programs

Maple

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

Mathematica

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}]

Formula

a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-2} binomial(n,k)^2 * (n-k) * a(k).