A156730 Triangle T(n, k) = Product_{j=1..k} Product_{i=0..j-1} ( 1 - (n-k+1)*(3*i-2) ) with T(n, 0) = 1 and T(n, n) = n!, read by rows.

1, 1, 1, 1, 5, 2, 1, 7, -25, 6, 1, 9, -98, -875, 24, 1, 11, -243, -15092, 398125, 120, 1, 13, -484, -98415, 46483360, 3441790625, 720, 1, 15, -845, -404624, 1076168025, 4151893715200, -743856998828125, 5040, 1, 17, -1350, -1263275, 11501032576, 458947996781625, -14092191572383232000, -4983748910023583984375, 40320

Offset

0,5

Comments

Row sums are: 1, 2, 8, -11, -939, 382922, 3488175820, -739704029345313, -4997840642636470626461, ...

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)*(3*i -2) ) 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) = (-3*(n-k+1))^binomial(k+1, 2)*Product_{j=1..k} Pochhammer( -(2*(n-k) + 3)/(3*(n-k+1)), j) with T(n, 0) = 1 and T(n, n) = n!. - G. C. Greubel, Feb 25 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  5,    2;
  1,  7,  -25,       6;
  1,  9,  -98,    -875,         24;
  1, 11, -243,  -15092,     398125,           120;
  1, 13, -484,  -98415,   46483360,    3441790625,              720;
  1, 15, -845, -404624, 1076168025, 4151893715200, -743856998828125, 5040;

Mathematica

(* First program *)
t[n_, k_]= If[k==0, n!, Product[1 -(3*i-2)*(k+1), {j, n}, {i, 0, j-1}]];
Table[t[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)
(* Second program *)
T[n_, k_, p_, q_]:= If[k==0, 1, If[k==n, n!, (-p*(n-k+1))^Binomial[k+1, 2]*Product[ Pochhammer[(q*(n-k+1) -1)/(p*(n-k+1)), j], {j, k}]]];
Table[T[n, k, 3, -2], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

Prog

(Sage)
@CachedFunction
def T(n, k, p, q):
    if (k==0): return 1
    elif (k==n): return factorial(n)
    else: return (-p*(n-k+1))^binomial(k+1, 2)*product( rising_factorial( (q*(n-k+1)-1)/(p*(n-k+1)), j) for j in (1..k))
flatten([[T(n, k, 3, -2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 24 2021
(Magma)
function T(n, k, p, q)
  if k eq 0 then return 1;
  elif k eq n then return Factorial(n);
  else return (&*[1 - (n-k+1)*(p*m+q): m in [0..j-1], j in [1..k]]);
  end if; return T;
end function;
[T(n, k, 3, -2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

Crossrefs

Cf. A156693, A156699, A156722, A156725, A156727.

Sequence in context: A111395 A188596 A199622 * A159549 A011394 A321289

Adjacent sequences: A156727 A156728 A156729 * A156731 A156732 A156733

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

Edited by G. C. Greubel, Feb 25 2021

Status

approved