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

0,3

Comments

Exponential transform of A000010.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..552

N. J. A. Sloane, Transforms

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

Example

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! + ...

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}]

Prog

(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

Crossrefs

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

Sequence in context: A187009 A144168 A177482 * A242154 A108124 A186365

Adjacent sequences: A300008 A300009 A300010 * A300012 A300013 A300014

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 09 2018

Status

approved