A156696 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 = 1, read by rows.

1, 1, 1, 1, -1, 1, 1, 5, 5, 1, 1, -45, 225, -45, 1, 1, 585, 26325, 26325, 585, 1, 1, -9945, 5817825, -52360425, 5817825, -9945, 1, 1, 208845, 2076963525, 243004732425, 243004732425, 2076963525, 208845, 1, 1, -5221125, 1090405850625, -2168817236893125, 28194624079610625, -2168817236893125, 1090405850625, -5221125, 1

Offset

0,8

Comments

Row sums are: {1, 2, 1, 12, 137, 53822, -40744663, 490163809592, 23859170407083377, 14660989220762621919002, -54998077449004520067705092623, ...}.

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 = 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, 2, -1). - G. C. Greubel, Feb 25 2021

Example

Triangle begins as:
  1;
  1,      1;
  1,     -1,          1;
  1,      5,          5,            1;
  1,    -45,        225,          -45,            1;
  1,    585,      26325,        26325,          585,          1;
  1,  -9945,    5817825,    -52360425,      5817825,      -9945,      1;
  1, 208845, 2076963525, 243004732425, 243004732425, 2076963525, 208845, 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, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 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, 2, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 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, 2, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 25 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, 2, -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A156697 (m=2), A156698 (m=3).

cf. A156690, A156691, A156692, A156693.

Cf. A156690, A156699, A156722.

Sequence in context: A188587 A373434 A174119 * A232651 A145227 A236555

Adjacent sequences: A156693 A156694 A156695 * A156697 A156698 A156699

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 13 2009

Extensions

Edited by G. C. Greubel, Feb 25 2021

Status

approved