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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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%I #16 Sep 08 2022 08:45:29

%S 1,3,4,-30,-216,420,14400,22680,-1411200,-8482320,195955200,

%T 2399997600,-36883123200,-788107320000,9066542284800,318173519664000,

%U -2824576634880000,-159078423407904000,1088403529973760000,97970873094110016000,-508476519708917760000,-73631427647097640320000

%N 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).

%D V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

%H G. C. Greubel, <a href="/A127068/b127068.txt">Table of n, a(n) for n = 0..150</a>

%F From _Peter Bala_, Feb 15 2022: (Start)

%F Conjectures:

%F a(2*n) = (-1)^(n+1)*(n + 1)*(2*n - 1)*(2*n)!.

%F 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)

%p T:= proc(n, k) option remember;

%p if k=0 then 1

%p elif k=1 then n

%p else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)

%p fi; end:

%p seq(T(3, n), n=0..25); # _G. C. Greubel_, Jan 29 2020

%t 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 *)

%o (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) ));

%o vector(25, n, T(3, (n-1)) ) \\ _G. C. Greubel_, Jan 29 2020

%o (Magma)

%o function T(n, k)

%o if k eq 0 then return 1;

%o elif k eq 1 then return n;

%o else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);

%o end if; return T; end function;

%o [T(3, n): n in [0..25]]; // _G. C. Greubel_, Jan 29 2020

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k==0): return 1

%o elif (k==1): return n

%o else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)

%o [T(3, n) for n in (0..25)] # _G. C. Greubel_, Jan 29 2020

%Y A column of A105937.

%K sign

%O 0,2

%A Vincent v.d. Noort, Mar 21 2007