OFFSET
0,2
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
a(n) = n!*Sum_{k=0..n} A000984(k)/(n-k)!. - Vladimir Kruchinin, Sep 10 2010
a(n) = Sum_{k=0..n} binomial(n,k)*(2*k)!/k!. - Vladimir Kruchinin, Sep 10 2010
a(n) ~ sqrt(2)*4^n*n^n/exp(n-1/4). - Vaclav Kotesovec, Jun 27 2013
D-finite with recurrence: a(n) - (4*n-1)*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Sep 27 2013
a(n) = U(1/2, n+3/2, 1/4)/2 where U denotes the Kummer U function. - Peter Luschny, Nov 26 2014
From Peter Bala, Nov 21 2017: (Start)
a(n+k) = a(n) (mod k) for all n and k. It follows that the sequence a(n) taken modulo k is periodic with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 3, 7, 3, 1, 1, 3, 7, 3, 1, ... with exact period 5.
The e.g.f. A(x) = 1/sqrt(1 - 4*x)*exp(x) satisfies the differential equation (1 - 4*x)A' - (3 - 4*x)*A = 0 with A(0) = 1. Mathar's recurrence above follows from this. (End)
MAPLE
A052143 := n -> KummerU(1/2, n+3/2, 1/4)/2:
seq(simplify(A052143(n)), n=0..17); # Peter Luschny, Dec 18 2017
MATHEMATICA
CoefficientList[Series[E^x/Sqrt[1-4*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
PROG
(Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)*(k)!, k, 0, n), n, 0, 12); /* Emanuele Munarini, Dec 17 2017 */
(PARI) x='x+O('x^99); Vec(serlaplace(exp(x)/sqrt(1-4*x))) \\ Altug Alkan, Dec 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 23 2000
STATUS
approved