A305619 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)).
0, 1, 2, -3, 4, -10, 636, -1078, -18416, -131976, 5035920, 5333592, 187347744, -4079616528, -14669908512, -140154110640, 28743506893056, -92449999037568, 2738959517576448, -52969092379214976, 34211286306178560, -16812071564735736576, 1407763084021569335808
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x^2/2! + 2*x^3/3! - 3*x^4/4! + 4*x^5/5! - 10*x^6/6! + ... exp(A(x)) = 1 + x^2/2 + x^3/3 + x^5/5 + x^7/7 + ... + x^A000040(k)/A000040(k) + ... exp(exp(A(x))-1) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + ... + A218002(k)*x^k/k! + ...
Programs
-
Maple
a:=series(log(1+add(x^ithprime(k)/ithprime(k),k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
-
Mathematica
nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^Prime[k]/Prime[k], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!] a[n_] := a[n] = Boole[PrimeQ[n]] (n - 1)! - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]