OFFSET
0,2
COMMENTS
|a(n)| is the number of functions f:{1,2,...,2n}->{1,2,...,2n} such that each element has either 0 or 2 preimages. That is, |(f^-1)(x)| is in {0,2} for all x in {1,2,...,2n}. - Geoffrey Critzer, Feb 24 2012.
REFERENCES
V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..150
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 131.
S. Goodenough, C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014-2016.
S. Goodenough, C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16.
FORMULA
E.g.f. for positive values with interpolated zeros:
(1-2*x^2)^(-1/2) which is exp(log(1/(1-x*G(x)))) where
G(x) is the e.g.f. for A036770. - Geoffrey Critzer, Feb 24 2012
a(n) = (-8)^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017
MAPLE
T:= proc(n, k) option remember;
if k=0 then 1
elif k=1 then n
else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
fi; end:
seq(T(0, 2*k), n=0..15); # G. C. Greubel, Jan 28 2020
MATHEMATICA
nn=40; b=(1-(1-2x^2)^(1/2))/x; Select[Range[0, nn]!CoefficientList[Series[1/(1-x b), {x, 0, nn}], x], #>0&]*Table[(-1)^(n), {n, 0, nn/2}] (* Geoffrey Critzer, Feb 24 2012 *)
T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, n, (n-k+1)*T[n+1, k-1] - (k-1)*(n+1)* T[n+2, k-2]]]; Table[T[0, 2*n], {n, 0, 15}] (* G. C. Greubel, Jan 28 2020 *)
PROG
(PARI) T(n, k) = if(k==0, 1, if(k==1, n, (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2) ));
vector(15, n, T(0, 2*(n-1)) ) \\ G. C. Greubel, Jan 28 2020
(Magma)
function T(n, k)
if k eq 0 then return 1;
elif k eq 1 then return n;
else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
end if; return T; end function;
[T(0, 2*n): n in [0..15]]; // G. C. Greubel, Jan 28 2020
(Sage)
@CachedFunction
def T(n, k):
if (k==0): return 1
elif (k==1): return n
else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
[T(0, 2*n) for n in (0..15)] # G. C. Greubel, Jan 28 2020
CROSSREFS
KEYWORD
sign
AUTHOR
Vincent v.d. Noort, Mar 21 2007
STATUS
approved