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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

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

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

Programs

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 *)

PARI

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

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

Extensions

More terms from Vladeta Jovovic, Jul 17 2006