A156699 Triangle T(n, k) = Product_{j=1..k} Product_{i=0..j-1} ( 1 - (n-k+1)*(2*i-1) ) with T(n, 0) = 1 and T(n, n) = n!, read by rows.

1, 1, 1, 1, 3, 2, 1, 4, -9, 6, 1, 5, -32, -135, 24, 1, 6, -75, -2048, 18225, 120, 1, 7, -144, -12375, 1835008, 31984875, 720, 1, 8, -245, -48384, 38795625, 32883343360, -954268745625, 5040, 1, 9, -384, -145775, 390168576, 3283855678125, -15321007338291200, -597882768540159375, 40320

Offset

0,5

Comments

Row sums are: 1, 2, 6, 2, -137, 16229, 33808092, -921346650220, -613200491632709703, 9136424641471148255125435, ...

Links

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Formula

Let the square array t(n, k) be given by t(n, k) = Product_{j=1..n} Product_{i=0..j-1} ( 1 - (k+1)*(2*i -1) ) with t(n, 0) = n!. The number triangle, T(n, k), is the downward antidiagonals, i.e. T(n, k) = t(k, n-k).

T(n, k) = (-2*(n-k+1))^binomial(k, 2)*(n-k+2)^k*Product_{j=1..k} Pochhammer( (n-k)/(2*(n-k+1)), j-1) with T(n, 0) = 1 and T(n, n) = n!. - G. C. Greubel, Feb 25 2021

Example

Triangle begins as:
  1;
  1, 1;
  1, 3,    2;
  1, 4,   -9,      6;
  1, 5,  -32,   -135,       24;
  1, 6,  -75,  -2048,    18225,         120;
  1, 7, -144, -12375,  1835008,    31984875,           720;
  1, 8, -245, -48384, 38795625, 32883343360, -954268745625, 5040;

Mathematica

(* First program *)
t[n_, k_]:= If[k==0, n!, Product[1 -(2*i-1)*(k+1), {j, n}, {i, 0, j-1}]];
Table[t[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)
(* Second program *)
T[n_, k_] = If[k==0, 1, If[k==n, n!, (-2*(n-k+1))^Binomial[k, 2]*(n-k+2)^k *Product[Pochhammer[(n-k)/(2*(n-k+1)), j-1], {j, k}] ]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

Prog

(Sage)
@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==n): return factorial(n)
    else: return (-2*(n-k+1))^binomial(k, 2)*(n-k+2)^k*product( rising_factorial( (n-k)/(2*(n-k+1)), j-1) for j in (1..k))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 24 2021
(Magma)
function T(n, k)
  if k eq 0 then return 1;
  elif k eq n then return Factorial(n);
  else return (&*[ (&*[1 - (n-k+1)*(2*m-1): m in [0..j-1]]) :j in [1..k]]);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

Crossrefs

Cf. A156698, A156730.

Sequence in context: A159966 A119263 A028412 * A245183 A262347 A182236

Adjacent sequences: A156696 A156697 A156698 * A156700 A156701 A156702

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 13 2009

Extensions

Edited by G. C. Greubel, Feb 25 2021

Status

approved