

A006040


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


13



1, 2, 9, 82, 1313, 32826, 1181737, 57905114, 3705927297, 300180111058, 30018011105801, 3632179343801922, 523033825507476769, 88392716510763573962, 17324972436109660496553, 3898118798124673611724426, 997918412319916444601453057, 288398421160455852489819933474
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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, 787805. (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(n1) + 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))/(1x) = 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)  (n1)^2*a(n1), n >= 2. The sequence b(n) := (n1)!^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  ...(n1)^2/(n^2+1))))). Hence BesselI(0,2) := sum {k = 0..inf} 1/k!^2 = 1 + 1/(1  1/(5  4/(10  ...(n1)^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[n1] + 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



