A001569 Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*(1-exp(x))^(1/2)). (Formerly M2161 N0861)

1, -1, -1, 2, 37, 329, 1501, -31354, -1451967, -39284461, -737652869, 560823394, 1103386777549, 82520245792997, 4398448305245905, 168910341581721494, 998428794798272641, -720450682719825322809, -105099789680808769094057, -10594247095804692725600734

Offset

0,4

References

S. M. Kerawala, Asymptotic solution of the "Probleme des menages", Bull. Calcutta Math. Soc., 39 (1947), 82-84.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Table of n, a(n) for n=0..19.

S. M. Kerawala, Asymptotic solution of the "Probleme des menages, Bull. Calcutta Math. Soc., 39 (1947), 82-84. [Annotated scanned copy]

Formula

Let b(n) satisfy (n-2)*b(n) - n*(n-2)*b(n-1) - n*b(n-2) = 0; write b(n) = (n!/e^2)*(1 + Sum_{r>=1} a_r/n^r).

a(n) = n!*Sum_{k=0..n} (-1)^k*Stirling2(n,k)/k!. - Vladeta Jovovic, Jul 17 2006

E.g.f.: 1 + x*(1 - E(0))/(1-x) where E(k) = 1 + 1/(1-x*(k+1))/(k+1)/(1-x/(x-1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 19 2013

E.g.f.: 1 + x*(1 - S)/(1-x) where S = Sum_{k>=0} (1 + 1/(1-x-x*k)/(k+1)) * x^k / Product_{i=0..k-1} (1-x-x*i)*(i+1). - Sergei N. Gladkovskii, Jan 21 2013

Mathematica

m = 20;
B[x_] = BesselI[0, x] + O[x]^(2 m) // Normal;
A[x_] = B[2(1 - E^x)^(1/2)] + O[x]^m;
CoefficientList[A[x], x]*Range[0, m-1]!^2 (* Jean-François Alcover, Oct 26 2019 *)

Prog

(PARI) a(n)=n!*sum(k=0, n, (-1)^k*stirling(n, k, 2)/k!) \\ Charles R Greathouse IV, Apr 18 2016

Crossrefs

Sequence in context: A078976 A200911 A243101 * A092853 A297796 A300542

Adjacent sequences: A001566 A001567 A001568 * A001570 A001571 A001572

Keyword

sign,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jul 17 2006

Status

approved