OFFSET
0,5
COMMENTS
Row sums are: {1, 2, -1, 92, 15797, 38476622, -954273578893, 258530641552220312, 1151446961763256010682377, ...}.
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 = 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, 3, -2). - G. C. Greubel, Feb 26 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, -3, 1;
1, 45, 45, 1;
1, -1215, 18225, -1215, 1;
1, 47385, 19190925, 19190925, 47385, 1;
1, -2416635, 38170749825, -1030610245275, 38170749825, -2416635, 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, 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, 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, 3, 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, 3, 3, -2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 26 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Feb 14 2009
EXTENSIONS
Edited by G. C. Greubel, Feb 26 2021
STATUS
approved