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

%I A337166 #9 Aug 19 2022 04:37:58
%S A337166 1,0,-1,-1,17,99,-926,-20385,25969,7206059,90298826,-3271747557,
%T A337166 -149187119280,236884125841,233237751740057,7110791842650002,
%U A337166 -293292401726383791,-32980038867059802549,-498084376275585698222,114298048468067933019627,9072219653673352772098960
%N A337166 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 + x - BesselI(0,2*sqrt(x))).
%F A337166 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-2} binomial(n,k)^2 * (n-k) * a(k).
%p A337166 A337166 := proc(n)
%p A337166     option remember ;
%p A337166     if n = 0 then
%p A337166         1;
%p A337166     else
%p A337166         add(binomial(n,k)^2*(n-k)*procname(k),k=0..n-2) ;
%p A337166         -%/n ;
%p A337166     end if;
%p A337166     simplify(%) ;
%p A337166 end proc:
%p A337166 seq(A337166(n),n=0..40) ; # _R. J. Mathar_, Aug 19 2022
%t A337166 nmax = 20; CoefficientList[Series[Exp[1 + x - BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
%t A337166 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}]
%Y A337166 Cf. A061696, A293037, A336209.
%K A337166 sign
%O A337166 0,5
%A A337166 _Ilya Gutkovskiy_, Jan 28 2021