A006040 a(n) = Sum_{i=0..n} (n!/(n-i)!)^2. (Formerly M1950)

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

Offset

0,2

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

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

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 *)

Prog

(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

Crossrefs

Main diagonal of array A099597.

Cf. A073701.

Sequence in context: A117581 A110567 A123570 * A067309 A087798 A113146

Adjacent sequences: A006037 A006038 A006039 * A006041 A006042 A006043

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

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

Status

approved