login
A126934
Define an array by d(m, 0) = 1, d(m, 1) = m; d(m, k) = (m - k + 1) d(m+1, k-1) - (k-1) (m+1) d(m+2, k-2). Sequence gives d(0,2n).
2
1, -2, 36, -1800, 176400, -28576800, 6915585600, -2337467932800, 1051860569760000, -607975409321280000, 438958245529964160000, -387161172557428389120000, 409616520565759235688960000, -512020650707199044611200000000, 746526108731096207043129600000000, -1255656914885703820246543987200000000
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
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
a(n) = (-1)^n * A001147(n) * A001813(n). - N. J. A. Sloane, Mar 21 2007
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
See A105937 for the full array.
See also A127080.
Sequence in context: A209803 A088026 A174881 * A303503 A178949 A200571
KEYWORD
sign
AUTHOR
Vincent v.d. Noort, Mar 21 2007
STATUS
approved