OFFSET
0,3
LINKS
Marko R. Riedel, Brian M. Scott et al. Enumerating a type of function by inclusion-exclusion
FORMULA
a(n) = Sum_{q=0..n} C(n,q) (-1)^q 2^q C(2n-q,n-q) (n-q)!.
a(n) = abs(A000806(n)) * 2^n.
E.g.f.: exp(-1+sqrt(1-4*x))/sqrt(1-4*x). - Benedict W. J. Irwin, Jul 14 2016
a(n) ~ 2^(2*n+1/2) * n^n / exp(n+1). - Vaclav Kotesovec, Jul 16 2016
Conjecture: Alternating sign g.f. is Sum_{k>=0} HermiteH[k,sqrt(x)]x^(k/2). - Benedict W. J. Irwin, Nov 30 2016
Conjecture D-finite with recurrence: a(n) + 2*(-2*n+1)*a(n-1) - 4*a(n-2)=0. - R. J. Mathar, Jan 27 2020
a(n) = KummerU(-n, -2*n, -2). - Peter Luschny, May 10 2022
MAPLE
a := n -> add(binomial(n, q)*(-1)^q*2^q*binomial(2*n-q, n-q)*(n-q)!, q=0..n): seq(a(n), n=0..20);
seq(simplify(KummerU(-n, -2*n, -2)), n = 0..18); # Peter Luschny, May 10 2022
MATHEMATICA
Table[CoefficientList[Series[E^(-1 + Sqrt[1 - 4 x])/Sqrt[1 - 4 x], {x, 0, 20}], x][[n]] (n - 1)!, {n, 1, 20}] (* Benedict W. J. Irwin, Jul 14 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Marko Riedel, Apr 23 2016
STATUS
approved