A300011 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k!), where phi() is the Euler totient function (A000010).

1, 1, 2, 6, 20, 80, 362, 1820, 10084, 60522, 391864, 2714514, 20001700, 156107224, 1284705246, 11112088358, 100698613720, 953478331288, 9410963022318, 96614921664444, 1029705968813656, 11373102766644372, 129972789566984682, 1534638410054873892, 18696544357738885720

Offset: 0

Author

Ilya Gutkovskiy, Mar 09 2018

Keywords

• nonn

Comments

Exponential transform of A000010.

Examples

			E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 80*x^5/5! + 362*x^6/6! + 1820*x^7/7! + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..552
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000010, A050392, A159929, A274804, A295739.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*numtheory[phi](j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 09 2018

Mathematica

nmax = 24; CoefficientList[Series[Exp[Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[EulerPhi[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

PARI

a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} A000010(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} phi(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022