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A262843 Inverse Moebius transform of n^(n-1). 13

%I #16 Jun 19 2022 15:23:16

%S 1,3,10,67,626,7788,117650,2097219,43046731,1000000628,25937424602,

%T 743008378540,23298085122482,793714773371796,29192926025391260,

%U 1152921504608944195,48661191875666868482,2185911559738739586477,104127350297911241532842,5242880000000001000000692,278218429446951548637314060

%N Inverse Moebius transform of n^(n-1).

%C Logarithmic derivative of A262842.

%H G. C. Greubel, <a href="/A262843/b262843.txt">Table of n, a(n) for n = 1..385</a>

%F a(n) = Sum{d|n} d^(d-1).

%F G.f.: Sum_{n>=1} n^(n-1) * x^n/(1 - x^n).

%e O.g.f.: A(x) = x + 3*x^2 + 10*x^3 + 67*x^4 + 626*x^5 + 7788*x^6 +...

%e where

%e A(x) = x/(1-x) + 2*x^2/(1-x^2) + 3^2*x^3/(1-x^3) + 4^3*x^4/(1-x^4) + 5^4*x^5/(1-x^5) + 6^5*x^6/(1-x^6) +...+ n^(n-1)* x^n/(1 -x^n) +...

%e Logarithmic generating function.

%e L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 67*x^4/4 + 626*x^5/5 + 7788*x^6/6 +...

%e where

%e exp(L(x)) = 1/( (1-x) * (1-x^2) * (1-x^3)^3 * (1-x^4)^16 * (1-x^5)^125 * (1-x^6)^1296 *...* (1-x^n)^(n^(n-2)) *...).

%e Explicitly,

%e exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 22*x^4 + 150*x^5 + 1469*x^6 + 18452*x^7 + 282426*x^8 +...+ A262842(n)*x^n ...

%t a[n_] := DivisorSum[n, #^(#-1) &]; Array[a, 30] (* _Jean-François Alcover_, Dec 23 2015 *)

%o (PARI) {a(n)=sumdiv(n,d, d^(d-1))}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) {a(n)=polcoeff(sum(m=1, n, m^(m-1)*x^m/(1-x^m +x*O(x^n))), n)}

%o for(n=1,30,print1(a(n),", "))

%o (Python)

%o from sympy import divisors

%o def A262843(n): return sum(d**(d-1) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 19 2022

%Y Cf. A262842, A062796.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 03 2015

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Last modified March 29 06:34 EDT 2024. Contains 371265 sequences. (Running on oeis4.)