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A131965 a(n) = 1 + Sum_{i=2..n-1} n*a(i).

%I A131965 #37 Sep 08 2022 08:45:31
%S A131965 1,1,1,4,21,131,943,7701,70409,712891,7921011,95844233,1254688141,
%T A131965 17670191319,266412115271,4281623281141,73073037331473,
%U A131965 1319881736799731,25155393101359579,504505383866156001,10621165976129600021,234196709773657680463,5397676549069062730671
%N A131965 a(n) = 1 + Sum_{i=2..n-1} n*a(i).
%C A131965 a(n) = 1 + Sum_{i=2..n-1} 1*a(i) = 2^n; a(n) = 1 + Sum_{i=2..n-1} 2*a(i) = 3^n; etc. It seems that a(n+1)/(n*a(n)) -> 1 for n -> oo. [Comment corrected by _Emeric Deutsch_, Aug 10 2007]
%C A131965 Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 4, 5, etc., along the main diagonal, and zeros everywhere else. Then a(n) equals the permanent of M(n-2) for n >= 3. - _John M. Campbell_, Apr 20 2021
%H A131965 Alois P. Heinz, <a href="/A131965/b131965.txt">Table of n, a(n) for n = 0..449</a>
%F A131965 a(n) = 1 + Sum_{i=2..n-1} n*a(i).
%F A131965 E.g.f.: 1/2 * (x + (2*exp(x)-5)/(x-1)^2 -5/(x-1)).
%F A131965 Asymptotic expansion: a(n)/n! = (5/2 + e)*n^2 + O(n).
%F A131965 a(n) = (n+1)*a(n-1) + a(n-2) + ... + a(2), e.g., a(5) = 6*21 + 4 + 1 = 131.
%e A131965 a(4)=21 because 1 + 4*1 + 4*4 = 21.
%p A131965 rctlnn := proc(n::nonnegint) local j; option remember; if n = 0 then 0; else 1+add(n*procname(j), j=2..n-1); end if; end proc:
%p A131965 a[1] := 1; for n from 2 to 18 do a[n] := 1+sum(n*a[i], i = 2 .. n-1) end do: seq(a[n], n = 1 .. 18); # _Emeric Deutsch_, Aug 10 2007
%p A131965 # third Maple program:
%p A131965 a:= proc(n) option remember;
%p A131965       1+add(n*a(i), i=2..n-1)
%p A131965     end:
%p A131965 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 03 2020
%t A131965 a[1] = a[2] = 1; a[n_] := a[n] = (n^2*a[n-1]-1)/(n-1); Array[a, 30] (* _Jean-François Alcover_, Feb 08 2017 *)
%o A131965 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (-8*(1+x) + 2*(3-x)*Exp(x) + (4+3*x^2-x^3))/(2*(1-x)^3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Mar 09 2019
%o A131965 (Sage) m = 25; T = taylor((-8*(1+x) + 2*(3-x)*exp(x) + (4+3*x^2-x^3))/(2*(1-x)^3), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Mar 09 2019
%Y A131965 Cf. A131407, A131408, A079750.
%K A131965 nonn
%O A131965 0,4
%A A131965 _Thomas Wieder_, Aug 02 2007
%E A131965 More terms from _Emeric Deutsch_, Aug 10 2007
%E A131965 a(0)=1 prepended and edited by _Alois P. Heinz_, Sep 03 2020