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URL: https://oeis.org/A227550

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A227550
A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.
6
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, 52, 154, 720, 5040, 874, 206, 88, 88, 206, 874, 5040, 40320, 5914, 1080, 294, 176, 294, 1080, 5914, 40320, 362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880, 3628800
OFFSET
0,4
COMMENTS
A003422 gives the second column (after 0).
LINKS
Vincenzo Librandi, Rows n = 0..70, flattened
FORMULA
From G. C. Greubel, May 02 2021: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(n, n) = n!.
Sum_{k=0..n} T(n, k) = 2^n * (1 +Sum_{j=1..n-1} j*j!/2^j) = A140710(n). (End)
EXAMPLE
Triangle begins:
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, 52, 154, 720;
5040, 874, 206, 88, 88, 206, 874, 5040;
40320, 5914, 1080, 294, 176, 294, 1080, 5914, 40320;
362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880;
MATHEMATICA
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]
PROG
(Haskell)
a227550 n k = a227550_tabl !! n !! k
a227550_row n = a227550_tabl !! n
a227550_tabl = map fst $ iterate
(\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
([1], a001563_list)
-- Reinhard Zumkeller, Aug 05 2013
(Magma)
function T(n, k)
if k eq 0 or k eq n then return Factorial(n);
else return T(n-1, k-1) + T(n-1, k);
end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021
(SageMath)
def T(n, k): return factorial(n) if (k==0 or k==n) else T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021
CROSSREFS
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).
Cf. A003422.
Cf. A227791 (central terms), A001563, A074911.
Sequence in context: A119918 A084867 A194949 * A286384 A379010 A099259
KEYWORD
nonn,tabl
AUTHOR
Vincenzo Librandi, Aug 04 2013
STATUS
approved