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

0,4

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

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