A337824 - OEIS

%I #11 Jan 07 2024 01:49:30

%S 0,1,2,-15,16,2505,-60264,-606515,131316928,-4813100271,-339213768200,

%T 62401665573621,-2075963863814928,-745086903175541927,

%U 140250562903680456332,808225064553580739325,-5491409141464496462591744,1013058261721909845376508449,127689148764914765889971316600

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

%H Robert Israel, <a href="/A337824/b337824.txt">Table of n, a(n) for n = 0..257</a>

%F Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + x * BesselI(0,2*sqrt(x))).

%F Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n^2 * x^n / (n!)^2).

%p S:= series(log(1+x*BesselI(0,2*sqrt(x))),x,31):

%p 0,seq(coeff(S,x,n)*(n!)^2, n=1..30); # _Robert Israel_, Jan 07 2024

%t a[0] = 0; a[n_] := a[n] = n^2 - (1/n) * Sum[(Binomial[n, k] (n - k))^2 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]

%t nmax = 18; CoefficientList[Series[Log[1 + x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

%Y Cf. A002190, A033464, A101981, A337590, A337591, A337825.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Sep 24 2020