A336290 - OEIS

%I #4 Jul 17 2020 00:35:56

%S 1,1,5,44,628,12994,363548,13141974,593579712,32644440048,

%T 2141946861312,164937634714896,14703536203936512,1500149281670010048,

%U 173464224256287048576,22541427301008492798144,3267767649638304967827456,525055667919614566758512640

%N a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.

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

%t a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 17}]

%t nmax = 17; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

%t nmax = 17; Assuming[x > 0, CoefficientList[Series[Exp[Exp[x] (EulerGamma - ExpIntegralEi[-x] + Log[x])], {x, 0, nmax}], x]] Range[0, nmax]!^2

%Y Cf. A001008, A002805, A087761, A193161, A336289.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 16 2020