A156690 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)*(i+1) ) and m = 1, read by rows.

1, 1, 1, 1, -3, 1, 1, 15, 15, 1, 1, -105, 525, -105, 1, 1, 945, 33075, 33075, 945, 1, 1, -10395, 3274425, -22920975, 3274425, -10395, 1, 1, 135135, 468242775, 29499294825, 29499294825, 468242775, 135135, 1, 1, -2027025, 91307341125, -63275987399625, 569483886596625, -63275987399625, 91307341125, -2027025, 1

Offset

0,5

Comments

Row sums are: {1, 2, -1, 32, 317, 68042, -16392913, 59935345472, 443114522425577, 41952026212764267602, -11773681484663891313796273, ...}.

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)*(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, 1, 1). - G. C. Greubel, Feb 25 2021

Example

Triangle begins as:
  1;
  1,      1;
  1,     -3,         1;
  1,     15,        15,           1;
  1,   -105,       525,        -105,           1;
  1,    945,     33075,       33075,         945,         1;
  1, -10395,   3274425,   -22920975,     3274425,    -10395,      1;
  1, 135135, 468242775, 29499294825, 29499294825, 468242775, 135135, 1;

Mathematica

(* First program *)
t[n_, k_]:= If[k==0, n!, Product[1 -(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, 1, 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, 1, 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, 1, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A156691 (m=2), A156692 (m=3).

Cf. A156693, A156696, A156722.

Sequence in context: A176225 A173917 A174410 * A228900 A060325 A087987

Adjacent sequences: A156687 A156688 A156689 * A156691 A156692 A156693

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 13 2009

Extensions

Edited by G. C. Greubel, Feb 25 2021

Status

approved