A156698 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)*(2*i-1) ) and m = 3, read by rows.

1, 1, 1, 1, -3, 1, 1, 33, 33, 1, 1, -627, 6897, -627, 1, 1, 16929, 3538161, 3538161, 16929, 1, 1, -592515, 3343562145, -63527680755, 3343562145, -592515, 1, 1, 25478145, 5032061028225, 2581447307479425, 2581447307479425, 5032061028225, 25478145, 1

Offset

0,5

Comments

Row sums are: {1, 2, -1, 68, 5645, 7110182, -56841741493, 5172958787971592, 4953496772756652670937, ...}.

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)*(2*i-1) ) and m = 3.

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) = (3, 2, -1). - G. C. Greubel, Feb 26 2021

Example

Triangle begins as:
  1;
  1,       1;
  1,      -3,          1;
  1,      33,         33,            1;
  1,    -627,       6897,         -627,          1;
  1,   16929,    3538161,      3538161,      16929,       1;
  1, -592515, 3343562145, -63527680755, 3343562145, -592515, 1;

Mathematica

(* First program *)
t[n_, k_]:= If[k==0, n!, Product[1 -(2*i-1)*(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, 3], {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, 3, 2, -1], {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, 3, 2, -1) 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, 3, 2, -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 26 2021

Crossrefs

Cf. A007318 (m=0), A156696 (m=1), A156697 (m=2), this sequence (m=3).

Cf. A156690, A156691, A156692, A156693.

Cf. A156692, A156699, A156727.

Sequence in context: A215750 A126465 A077509 * A083993 A156727 A173505

Adjacent sequences: A156695 A156696 A156697 * A156699 A156700 A156701

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 13 2009

Extensions

Edited by G. C. Greubel, Feb 26 2021

Status

approved