A336290 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.

1, 1, 5, 44, 628, 12994, 363548, 13141974, 593579712, 32644440048, 2141946861312, 164937634714896, 14703536203936512, 1500149281670010048, 173464224256287048576, 22541427301008492798144, 3267767649638304967827456, 525055667919614566758512640

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

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

Mathematica

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}]
nmax = 17; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 17; Assuming[x > 0, CoefficientList[Series[Exp[Exp[x] (EulerGamma - ExpIntegralEi[-x] + Log[x])], {x, 0, nmax}], x]] Range[0, nmax]!^2

Crossrefs

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

Sequence in context: A052803 A201923 A222059 * A214396 A252931 A229396

Adjacent sequences: A336287 A336288 A336289 * A336291 A336292 A336293

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 16 2020

Status

approved