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<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
(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
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