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 A131965 a(n) = 1 + Sum_{i=2..n-1} n*a(i). 1
 1, 1, 1, 4, 21, 131, 943, 7701, 70409, 712891, 7921011, 95844233, 1254688141, 17670191319, 266412115271, 4281623281141, 73073037331473, 1319881736799731, 25155393101359579, 504505383866156001, 10621165976129600021, 234196709773657680463, 5397676549069062730671 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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] 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..449 FORMULA a(n) = 1 + Sum_{i=2..n-1} n*a(i). E.g.f.: 1/2 * (x + (2*exp(x)-5)/(x-1)^2 -5/(x-1)). Asymptotic expansion: a(n)/n! = (5/2 + e)*n^2 + O(n). a(n) = (n+1)*a(n-1) + a(n-2) + ... + a(2), e.g., a(5) = 6*21 + 4 + 1 = 131. EXAMPLE a(4)=21 because 1 + 4*1 + 4*4 = 21. MAPLE 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: 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 # third Maple program: a:= proc(n) option remember;       1+add(n*a(i), i=2..n-1)     end: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 03 2020 MATHEMATICA 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 *) PROG (MAGMA) m:=25; R:=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 (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 CROSSREFS Cf. A131407, A131408, A079750. Sequence in context: A078591 A090366 A273956 * A332851 A303563 A284816 Adjacent sequences:  A131962 A131963 A131964 * A131966 A131967 A131968 KEYWORD nonn AUTHOR Thomas Wieder, Aug 02 2007 EXTENSIONS More terms from Emeric Deutsch, Aug 10 2007 a(0)=1 prepended and edited by Alois P. Heinz, Sep 03 2020 STATUS approved

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Last modified June 14 01:29 EDT 2021. Contains 345016 sequences. (Running on oeis4.)