A006040 - OEIS

%I M1950 #43 Oct 08 2023 04:52:40

%S 1,2,9,82,1313,32826,1181737,57905114,3705927297,300180111058,

%T 30018011105801,3632179343801922,523033825507476769,

%U 88392716510763573962,17324972436109660496553,3898118798124673611724426,997918412319916444601453057,288398421160455852489819933474

%N a(n) = Sum_{i=0..n} (n!/(n-i)!)^2.

%D R. K. Guy, personal communication.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A006040/b006040.txt">Table of n, a(n) for n = 0..250</a>

%H D. Deford, <a href="http://www.math.dartmouth.edu/~ddeford/graphs.pdf">Seating rearrangements on arbitrary graphs</a>, 2013. (See Table 1)

%H D. Deford, <a href="http://dx.doi.org/10.2140/involve.2014.7.787">Seating rearrangements on arbitrary graphs</a>, involve, Vol. 7 (2014), No. 6, 787-805. (See Table 1)

%H R. K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%F a(n) = n^2*a(n-1) + 1.

%F The following formulas will need adjusting, since I have changed the offset. - _N. J. A. Sloane_, Dec 17 2013

%F a(n+1) = Nearest integer to BesselI(0, 2)*n!*n!, n >= 1.

%F a(n+1) = n!^2*Sum_{k = 0..n} 1/k!^2. BesselI(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n+1)*x^n/n!^2. - _Vladeta Jovovic_, Aug 30 2002

%F Recurrence: a(1) = 1, a(2) = 2, a(n+1) = (n^2 + 1)*a(n) - (n - 1)^2*a(n-1), n >= 2. The sequence defined by b(n) := (n-1)!^2 satisfies the same recurrence with the initial conditions b(1) = 1, b(2) = 1. It follows that a(n+1) = n!^2*(1 + 1/(1 - 1/(5 - 4/(10 - ... - (n - 1)^2/(n^2 + 1))))). Hence BesselI(0,2) := Sum_{k >= 0} 1/k!^2 = 1 + 1/(1 - 1/(5 - 4/(10 - ... - (n - 1)^2/(n^2 + 1 - ...)))). Cf. A073701. - _Peter Bala_, Jul 09 2008

%p a[0]:= 1:

%p for n from 1 to 30 do a[n]:= n^2*a[n-1] + 1 od:

%p seq(a[i],i=0..30); # _Robert Israel_, Dec 15 2014

%t a = 1; lst = {a}; Do[a = a * n^2 + 1; AppendTo[lst, a], {n, 1, 14}]; lst (* _Zerinvary Lajos_, Jul 08 2009 *)

%t Table[Sum[(n!/(n - k)!)^2, {k, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Aug 15 2017 *)

%o (PARI) a(n)=sum(k=0, n, (k!*binomial(n, k))^2 ); \\ _Joerg Arndt_, Dec 14 2014

%o (Sage)

%o def A006040_list(len):

%o     L = [1]

%o     for k in range(1,len): L.append(L[-1]*k^2+1)

%o     return L

%o A006040_list(18) # _Peter Luschny_, Dec 15 2014

%Y Main diagonal of array A099597.

%Y Cf. A073701.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E Offset changed by _N. J. A. Sloane_, Dec 17 2013