A156725 Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 2, read by rows.

1, 1, 1, 1, -2, 1, 1, 22, 22, 1, 1, -440, 4840, -440, 1, 1, 12760, 2807200, 2807200, 12760, 1, 1, -484880, 3093534400, -61870688000, 3093534400, -484880, 1, 1, 22789360, 5525052438400, 3204530414272000, 3204530414272000, 5525052438400, 22789360, 1

Offset

0,5

Comments

Row sums are: {1, 2, 0, 46, 3962, 5639922, -55684588958, 6420110978999522, 8653645559546848833282, 120959123027642635275104364802, ...}.

Links

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

Formula

T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 2.

T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (2, 3, -2). - G. C. Greubel, Feb 26 2021

Example

Triangle begins as:
  1;
  1,       1;
  1,      -2,          1;
  1,      22,         22,            1;
  1,    -440,       4840,         -440,          1;
  1,   12760,    2807200,      2807200,      12760,       1;
  1, -484880, 3093534400, -61870688000, 3093534400, -484880, 1;

Mathematica

(* First program *)
t[n_, k_]:= If[k==0, n!, Product[1 -(3*i-2)*(k+1), {j, n}, {i, 0, j-1}] ];
T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 26 2021 *)
(* Second program *)
f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];
T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));
Table[T[n, k, 2, 3, -2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 26 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))
def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))
flatten([[T(n, k, 2, 3, -2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 26 2021
(Magma)
f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;
T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;
[T(n, k, 2, 3, -2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 26 2021

Crossrefs

Cf. A007318 (m=0), A156722 (m=1), this sequence (m=2), A156727 (m=3).

Cf. A156691, A156697, A156730.

Sequence in context: A174174 A156889 A172177 * A141904 A246072 A147802

Adjacent sequences: A156722 A156723 A156724 * A156726 A156727 A156728

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

Edited by G. C. Greubel, Feb 26 2021

Status

approved