|
%I #10 Sep 08 2022 08:45:17
%S 1,1,0,1,1,-2,1,2,-2,0,1,3,0,-12,36,1,4,4,-24,24,0,1,5,10,-30,-60,420,
%T -1800,1,6,18,-24,-216,720,-720,0,1,7,28,0,-420,420,5040,-30240,
%U 176400,1,8,40,48,-624,-960,14400,-40320,40320,0,1,9,54,126,-756,-3780,22680,22680,-589680,3764880,-28576800
%N Infinite square array read by antidiagonals: T(m, 0) = 1, T(m, 1) = m; T(m, k) = (m - k + 1) T(m+1, k-1) - (k-1) (m+1) T(m+2, k-2).
%D V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.
%H G. C. Greubel, <a href="/A105937/b105937.txt">Antidiagonals n = 0..100, flattened</a>
%F See A127080 for e.g.f.
%e Array begins
%e 1 1 1 1 1 1 1 1 1 1 ... (A000012)
%e 0 1 2 3 4 5 6 7 8 9 ... (A001477)
%e -2 -2 0 4 10 18 28 40 54 70 ... (A028552)
%e 0 12 24 30 24 0 48 126 240 396 ... (A126935)
%e 36 24 60 216 420 624 756 720 396 360 ... (A126958)
%e ...
%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(seq(T(n-k, k), k=0..n), n=0..12); # _G. C. Greubel_, Jan 28 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[n-k,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 28 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) )); \\ _G. C. Greubel_, Jan 28 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(n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 28 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(n-k, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 28 2020
%Y Rows give A000027, A028552, A126935, A126958.
%Y Columns give A126934, A126962, A127067, A127068, A127070.
%Y A127080 gives another version of the array.
%K sign,tabl
%O 0,6
%A Vincent v.d. Noort, Mar 24 2007
%E More terms added by _G. C. Greubel_, Jan 28 2020
|
