cp's OEIS Frontend

A127068 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).

%I A127068 #16 Sep 08 2022 08:45:29
%S A127068 1,3,4,-30,-216,420,14400,22680,-1411200,-8482320,195955200,
%T A127068 2399997600,-36883123200,-788107320000,9066542284800,318173519664000,
%U A127068 -2824576634880000,-159078423407904000,1088403529973760000,97970873094110016000,-508476519708917760000,-73631427647097640320000
%N A127068 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 A127068 V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.
%H A127068 G. C. Greubel, <a href="/A127068/b127068.txt">Table of n, a(n) for n = 0..150</a>
%F A127068 From _Peter Bala_, Feb 15 2022: (Start)
%F A127068 Conjectures:
%F A127068 a(2*n) = (-1)^(n+1)*(n + 1)*(2*n - 1)*(2*n)!.
%F A127068 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 A127068 T:= proc(n, k) option remember;
%p A127068       if k=0 then 1
%p A127068     elif k=1 then n
%p A127068     else (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
%p A127068       fi; end:
%p A127068 seq(T(3, n), n=0..25); # _G. C. Greubel_, Jan 29 2020
%t A127068 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 A127068 (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 A127068 vector(25, n, T(3, (n-1)) ) \\ _G. C. Greubel_, Jan 29 2020
%o A127068 (Magma)
%o A127068 function T(n, k)
%o A127068   if k eq 0 then return 1;
%o A127068   elif k eq 1 then return n;
%o A127068   else return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2);
%o A127068   end if; return T; end function;
%o A127068 [T(3, n): n in [0..25]]; // _G. C. Greubel_, Jan 29 2020
%o A127068 (Sage)
%o A127068 @CachedFunction
%o A127068 def T(n, k):
%o A127068     if (k==0): return 1
%o A127068     elif (k==1): return n
%o A127068     else: return (n-k+1)*T(n+1, k-1) - (k-1)*(n+1)*T(n+2, k-2)
%o A127068 [T(3, n) for n in (0..25)] # _G. C. Greubel_, Jan 29 2020
%Y A127068 A column of A105937.
%K A127068 sign
%O A127068 0,2
%A A127068 Vincent v.d. Noort, Mar 21 2007