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%I #24 May 03 2021 01:24:41
%S 1,1,1,2,2,2,6,4,4,6,24,10,8,10,24,120,34,18,18,34,120,720,154,52,36,
%T 52,154,720,5040,874,206,88,88,206,874,5040,40320,5914,1080,294,176,
%U 294,1080,5914,40320,362880,46234,6994,1374,470,470,1374,6994,46234,362880,3628800
%N A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.
%C A003422 gives the second column (after 0).
%H Vincenzo Librandi, <a href="/A227550/b227550.txt">Rows n = 0..70, flattened</a>
%F From _G. C. Greubel_, May 02 2021: (Start)
%F T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(n, n) = n!.
%F Sum_{k=0..n} T(n, k) = 2^n * (1 +Sum_{j=1..n-1} j*j!/2^j) = A140710(n). (End)
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 2, 2;
%e 6, 4, 4, 6;
%e 24, 10, 8, 10, 24;
%e 120, 34, 18, 18, 34, 120;
%e 720, 154, 52, 36, 52, 154, 720;
%e 5040, 874, 206, 88, 88, 206, 874, 5040;
%e 40320, 5914, 1080, 294, 176, 294, 1080, 5914, 40320;
%e 362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880;
%t t = {}; Do[r = {}; Do[If[k == 0||k == n, m = n!, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]
%o (Haskell)
%o a227550 n k = a227550_tabl !! n !! k
%o a227550_row n = a227550_tabl !! n
%o a227550_tabl = map fst $ iterate
%o (\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
%o ([1], a001563_list)
%o -- _Reinhard Zumkeller_, Aug 05 2013
%o (Magma)
%o function T(n,k)
%o if k eq 0 or k eq n then return Factorial(n);
%o else return T(n-1,k-1) + T(n-1,k);
%o end if; return T;
%o end function;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 02 2021
%o (Sage)
%o def T(n,k): return factorial(n) if (k==0 or k==n) else T(n-1, k-1) + T(n-1, k)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 02 2021
%Y Cf. similar triangles with t on the borders: A007318 (t = 1), A028326 (t = 2), A051599 (t = prime(n)), A051601 (t = n), A051666 (t = n^2), A108617 (t = fibonacci(n)), A134636 (t = 2n+1), A137688 (t = 2^n), A227075 (t = 3^n).
%Y Cf. A003422.
%Y Cf. A227791 (central terms), A001563, A074911.
%K nonn,tabl
%O 0,4
%A _Vincenzo Librandi_, Aug 04 2013
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