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A127068
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Let d(m, 0) = 1, d(m, 1) = m, and d(m, k) = (m - k + 1)*d(m+1, k-1) - (k-1)*(m+1) d(m+2, k-2). Sequence gives d(3,n).
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2
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1, 3, 4, -30, -216, 420, 14400, 22680, -1411200, -8482320, 195955200, 2399997600, -36883123200, -788107320000, 9066542284800, 318173519664000, -2824576634880000, -159078423407904000, 1088403529973760000, 97970873094110016000, -508476519708917760000, -73631427647097640320000
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OFFSET
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0,2
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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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Conjectures:
a(2*n) = (-1)^(n+1)*(n + 1)*(2*n - 1)*(2*n)!.
a(2*n+1) = - 2*(2*n + 3)*(3*n - 2)*a(2*n-1) - 4*(n - 1)*(2*n + 3)*(4*n^2 - 1)*a(2*n-3) with a(1) = 3 and a(3) = -30. (End)
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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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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[3, n], {n, 0, 25}] (* G. C. Greubel, Jan 29 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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