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A126934
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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).
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2
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1, -2, 36, -1800, 176400, -28576800, 6915585600, -2337467932800, 1051860569760000, -607975409321280000, 438958245529964160000, -387161172557428389120000, 409616520565759235688960000, -512020650707199044611200000000, 746526108731096207043129600000000, -1255656914885703820246543987200000000
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OFFSET
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0,2
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COMMENTS
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|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.
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REFERENCES
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V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.
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LINKS
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FORMULA
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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
a(n) = (-8)^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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) ));
(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;
(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)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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Vincent v.d. Noort, Mar 21 2007
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STATUS
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approved
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