A140710 -id:A140710 - cp's OEIS frontend

A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

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

Vincenzo Librandi, Aug 04 2013

A003422 gives the second column (after 0).

			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;

Vincenzo Librandi, Rows n = 0..70, flattened

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.

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

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

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]

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

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)

A140709 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 5, 3, 1, 87, 20, 8, 4, 1, 567, 107, 28, 12, 5, 1, 4167, 674, 135, 40, 17, 6, 1, 34407, 4841, 809, 175, 57, 23, 7, 1, 316647, 39248, 5650, 984, 232, 80, 30, 8, 1, 3219687, 355895, 44898, 6634, 1216, 312, 110, 38, 9, 1, 35878887, 3575582, 400793, 51532, 7850, 1528, 422, 148, 47, 10, 1

Offset: 1

Emeric Deutsch, Jun 03 2008

A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

			T(2,1)=1 (the vertical domino); T(2,2)=1 (the horizontal domino); T(3,1)=3 because we have (3), (1,2) and (2,1,1), where (a,b,c,...) stands for a polyomino with columns of lengths a,b,c,..., starting at level 0.
Triangle starts:
    1;
    1,   1;
    3,   2,   1;
   15,   5,   3,   1;
   87,  20,   8,   4,   1;
  567, 107,  28,  12,   5,   1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Cf. A000142, A132371, A140710.

T:=proc(n,k) options operator, arrow: binomial(n-1, k-1)+sum(factorial(j)*(j-1)*binomial(n-1-j, k-1),j=2..n-1) end proc: for n to 11 do seq(T(n, k),k=1..n) end do; # yields sequence in triangular form

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==1, n! -Sum[j!, {j,n-1}], T[n-1, k] + T[n-1, k-1] ]];
Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, May 02 2021 *)

T(n,k) = binomial(n-1, k-1) + sum(j=2, n-1, j!*(j-1)*binomial(n-1-j, k-1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 16 2019

@CachedFunction
def T(n, k):
    if (k < 0 or k > n): return 0
    elif (k==1): return factorial(n) - sum(factorial(j) for j in (1..n-1))
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 02 2021

T(n,k) = binomial(n-1, k-1) + Sum_{j=2..n-1} j!*(j-1)*binomial(n-1-j, k-1).
T(n,k) = T(n-1, k) + T(n-1, k-1) for n,k >= 2.
Sum of entries in row n is n! (A000142).
T(n,1) = A132371(n).
Sum_{k=1..n} k*T(n,k) = A140710(n).

cp's OEIS Frontend

A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

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