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A221741 a(n) = -4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n).

%I A221741 #41 Apr 21 2023 13:00:40
%S A221741 1,5,9,97,373,7625,48913,1513361,13717421,570623341,6698798233,
%T A221741 350549891889,5057809205989,319164643134737,5465701947765793,
%U A221741 403925909124187873,8008631808527689309,678470389458269406421,15287592943577781017641
%N A221741 a(n) = -4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n).
%C A221741 Per exhaustive program, written for bases from 2 to 10, the number of permutations pairs, which have the same ratio, equal to A221740(n)/a(n) = (n^2 (n+1)^n-(n+1)^n+1) / (-n^2+n (n+1)^n+(n+1)^n-n-1), is: {2,2,3,3,5,3,7,5,7,...} for n>=1 where n=r-1 and r is the base radix. Judging by above sequence it appears that the number of such permutations pairs is related to phi, which is the Euler totient function - according to A039649, A039650, A214288 (see bullet 1 of the analysis in the answer section of the StackExchange link). _Alexander R. Povolotsky_, Jan 26 2013
%H A221741 G. C. Greubel, <a href="/A221741/b221741.txt">Table of n, a(n) for n = 1..385</a>
%H A221741 NMBRTHRY, <a href="http://old.nabble.com/Number-of-specific-permutations-pairs-relates-to-Euler-Phi-totient-function---to34951281.html">Number of specific permutations pairs relates to Euler Phi totient function ?</a>
%H A221741 StackExchange, <a href="http://math.stackexchange.com/questions/210578">Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0</a>.
%F A221741 a(n) = -4*A023811(n+1)/((-3 + (-1)^n)*n).
%t A221741 Table[-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), {n,1,50}] (* _G. C. Greubel_, Feb 19 2017 *)
%o A221741 (Maxima) makelist(-4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n), n, 1, 20); /* _Martin Ettl_, Jan 25 2013 */
%o A221741 (PARI) for(n=1,25, print1(-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), ", ")) \\ _G. C. Greubel_, Feb 19 2017
%Y A221741 Cf. A221740, A023811.
%K A221741 nonn,easy
%O A221741 1,2
%A A221741 _Alexander R. Povolotsky_, Jan 23 2013