A305618 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)!).
0, 1, 1, -3, -9, 20, 190, -126, -6280, -10326, 293041, 1519320, -16985045, -194560444, 1013712777, 27317463952, -19210030599, -4305097718760, -17733269020226, 743855089334604, 7868686621862292, -132351392654695270, -2854492900112993039, 20150897206881256464
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x^2/2! + x^3/3! - 3*x^4/4! - 9*x^5/5! + 20*x^6/6! + ... exp(A(x)) = 1 + x^2/2! + x^3/3! + x^5/5! + x^7/7! + ... + x^A000040(k)/A039716(k) + ... exp(exp(A(x))-1) = 1 + x^2/2! + x^3/3! + 3*x^4/4! + 11*x^5/5! + ... + A190476(k)*x^k/k! + ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..472
Programs
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Maple
a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*t(n-j)* j*binomial(n, j), j=1..n-1)/n))(i-> `if`(isprime(i), 1, 0)) end: seq(a(n), n=1..25); # Alois P. Heinz, Dec 04 2018 -
Mathematica
nmax = 24; 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]] - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 24}]
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