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

1, 1, 1, 4, 21, 131, 943, 7701, 70409, 712891, 7921011, 95844233, 1254688141, 17670191319, 266412115271, 4281623281141, 73073037331473, 1319881736799731, 25155393101359579, 504505383866156001, 10621165976129600021, 234196709773657680463, 5397676549069062730671

Offset: 0

Author

Thomas Wieder, Aug 02 2007

Keywords

• nonn

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

Examples

			a(4)=21 because 1 + 4*1 + 4*4 = 21.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A131407, A131408, A079750.

Programs

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

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 *)

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

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.

Extensions

More terms from Emeric Deutsch, Aug 10 2007

a(0)=1 prepended and edited by Alois P. Heinz, Sep 03 2020