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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(4,n).
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%I #8 Sep 08 2022 08:45:29

%S 1,4,10,-24,-420,-960,22680,201600,-1496880,-36288000,64864800,

%T 7823692800,25297272000,-2092278988800,-18988521552000,

%U 690452066304000,11457025515936000,-277436193914880000,-7430805000755136000,133809610449715200000,5500591866494524800000,-76432049488877322240000

%N 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(4,n).

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

%H G. C. Greubel, <a href="/A127070/b127070.txt">Table of n, a(n) for n = 0..150</a>

%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(4, 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[4, 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(4, (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(4, 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(4, 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