OFFSET
1,2
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.
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
David W. Wilson, Table of n, a(n) for n = 1..100
FORMULA
a(n) = coefficient of t^2 in polynomial p(t) = Sum_{k=0..n} 2*n*C(2n-k,k)*(n-k)!*(t-1)^k/(2n-k).
a(n) = Sum_{k=2..n} (-1)^k*n*(2n-k-1)!*(n-k)!/((2n-2k)!*(k-2)!). - David W. Wilson, Jun 22 2006
a(n) = n*A000426(n) - Vladeta Jovovic, Dec 27 2007
Recurrence: (n-3)*(n-2)*(2*n-5)*(2*n-7)*a(n) = (n-3)*(n-2)*n*(2*n-7)^2*a(n-1) + (n-4)*(n-3)*n*(2*n-3)^2*a(n-2) + (n-2)*n*(2*n-5)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 26 2012
a(n) ~ 2/e^2*n!. - Vaclav Kotesovec, Oct 26 2012
From Mark van Hoeij, Jun 09 2019: (Start)
a(n) = round(2*(exp(-2)*n*(4*BesselK(n,2)-(2*n-5)*BesselK(n-1,2))-(-1)^n)) for n > 9.
Conjecture: a(n) + 2*a(n+p) + a(n+2*p) is divisible by p for any prime p. - Mark van Hoeij, Jun 10 2019
MATHEMATICA
Table[n*Sum[(-1)^k*(2*n-k-1)!*(n-k)!/((2*n-2*k)!*(k-2)!), {k, 2, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 26 2012 *)
PROG
(Haskell)
fac = a000142
a n = sum $ map f [2..n]
where f k = g k `div` h k
g k = (-1)^k * n * fac (2*n-k-1) * fac (n-k)
h k = fac (2*n-2*k) * fac (k-2)
-- James Spahlinger, Oct 08 2012
(Magma) [0] cat [&+[(-1)^k*n*Factorial(2*n-k-1)*Factorial(n-k)/(Factorial(2*n-2*k)*Factorial(k-2)): k in [2..n]]: n in [2..25]]; // Vincenzo Librandi, Jun 11 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended to 34 terms by N. J. A. Sloane, May 25 2005
Edited and further extended by David W. Wilson, Dec 27 2007
STATUS
approved