A006040 a(n) = Sum_{i=0..n} (n!/(n-i)!)^2.

1, 2, 9, 82, 1313, 32826, 1181737, 57905114, 3705927297, 300180111058, 30018011105801, 3632179343801922, 523033825507476769, 88392716510763573962, 17324972436109660496553, 3898118798124673611724426, 997918412319916444601453057, 288398421160455852489819933474

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

References

• R. K. Guy, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• D. Deford, Seating rearrangements on arbitrary graphs, 2013. (See Table 1)
• D. Deford, Seating rearrangements on arbitrary graphs, involve, Vol. 7 (2014), No. 6, 787-805. (See Table 1)
• R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Main diagonal of array A099597.

Cf. A073701.

Programs

Maple

a[0]:= 1:
for n from 1 to 30 do a[n]:= n^2*a[n-1] + 1 od:
seq(a[i],i=0..30); # Robert Israel, Dec 15 2014

Mathematica

a = 1; lst = {a}; Do[a = a * n^2 + 1; AppendTo[lst, a], {n, 1, 14}]; lst (* Zerinvary Lajos, Jul 08 2009 *)
Table[Sum[(n!/(n - k)!)^2, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 15 2017 *)

PARI

a(n)=sum(k=0, n, (k!*binomial(n, k))^2 ); \\ Joerg Arndt, Dec 14 2014

Sage

def A006040_list(len):
    L = [1]
    for k in range(1,len): L.append(L[-1]*k^2+1)
    return L
A006040_list(18) # Peter Luschny, Dec 15 2014

Formula

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

The following formulas will need adjusting, since I have changed the offset. - N. J. A. Sloane, Dec 17 2013

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

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

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

Extensions

Offset changed by N. J. A. Sloane, Dec 17 2013