A156722 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 = 1, read by rows.

1, 1, 1, 1, -1, 1, 1, 7, 7, 1, 1, -91, 637, -91, 1, 1, 1729, 157339, 157339, 1729, 1, 1, -43225, 74736025, -971568325, 74736025, -43225, 1, 1, 1339975, 57920419375, 14306343585625, 14306343585625, 57920419375, 1339975, 1, 1, -49579075, 66434721023125, -410234402317796875, 7794453644038140625, -410234402317796875, 66434721023125, -49579075, 1

Offset

0,8

Comments

Row sums are: {1, 2, 1, 16, 457, 318138, -822182723, 28728530689952, 6974117708745434977, 19261978962188975367009202, ...}.

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 = 1.

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

Example

Triangle begins as:
  1;
  1,       1;
  1,      -1,           1;
  1,       7,           7,              1;
  1,     -91,         637,            -91,              1;
  1,    1729,      157339,         157339,           1729,           1;
  1,  -43225,    74736025,     -971568325,       74736025,      -43225,       1;
  1, 1339975, 57920419375, 14306343585625, 14306343585625, 57920419375, 1339975, 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, 1], {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, 1, 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, 1, 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, 1, 3, -2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 26 2021

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A156725 (m=2), A156727 (m=3).

Cf. A156690, A156696, A156730.

Sequence in context: A140136 A281123 A171707 * A152565 A174497 A146322

Adjacent sequences: A156719 A156720 A156721 * A156723 A156724 A156725

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 14 2009

Extensions

Edited by G. C. Greubel, Feb 26 2021

Status

approved