A174125 Triangle T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2, read by rows.

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 24, 45, 24, 1, 1, 35, 105, 105, 35, 1, 1, 48, 210, 336, 210, 48, 1, 1, 63, 378, 882, 882, 378, 63, 1, 1, 80, 630, 2016, 2940, 2016, 630, 80, 1, 1, 99, 990, 4158, 8316, 8316, 4158, 990, 99, 1, 1, 120, 1485, 7920, 20790, 28512, 20790, 7920, 1485, 120, 1

Offset

0,5

Comments

Triangles of this class, depending upon q, are of the form T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and have the row sums Sum_{k=0..n} T(n, k, q) = q*(q+1)*C_{n+q}/binomial(n+2*q, q-1) - 2*q + q*[n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. - G. C. Greubel, Feb 11 2021

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

Let c(n, q) = Product_{j=2..n} j*(j+q) for n > 2, otherwise 1, then the number triangle is given by T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) for q = 2.

From G. C. Greubel, Feb 11 2021: (Start)

T(n, k, q) = (q+1)*binomial(n, k)*(Pochhammer(q+1, n)/(Pochhammer(q+1, k)*Pochhammer(q+1, n-k))), with T(n, 0) = T(n, n) = 1, and q = 2.

Sum_{k=0..n} T(n, k, 1) = 6*A000108(n+2)/(n+4) - 4 + 2*[n=0]. (End)

Example

Triangle begins as:
  1;
  1,   1;
  1,   8,    1;
  1,  15,   15,    1;
  1,  24,   45,   24,     1;
  1,  35,  105,  105,    35,     1;
  1,  48,  210,  336,   210,    48,     1;
  1,  63,  378,  882,   882,   378,    63,    1;
  1,  80,  630, 2016,  2940,  2016,   630,   80,    1;
  1,  99,  990, 4158,  8316,  8316,  4158,  990,   99,   1;
  1, 120, 1485, 7920, 20790, 28512, 20790, 7920, 1485, 120, 1;

Mathematica

(* First program *)
c[n_, q_]:= If[n<2, 1, Product[i*(i+q), {i, 2, n}]];
T[n_, m_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
T[n_, k_, q_]:= If[k==0 || k==n, 1, (q+1)*Binomial[n, k]*(Pochhammer[q+1, n]/(Pochhammer[q+1, k]*Pochhammer[q+1, n-k]))];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)

Prog

(Sage)
def T(n, k, q): return 1 if (k==0 or k==n) else (q+1)*binomial(n, k)*(rising_factorial(q+1, n)/(rising_factorial(q+1, k)*rising_factorial(q+1, n-k)))
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021
(Magma)
c:= func< n, q | n lt 2 select 1 else (&*[j*(j+q): j in [2..n]]) >;
T:= func< n, k, q | c(n, q)/(c(k, q)*c(n-k, q)) >;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

Crossrefs

Cf. A174124 (q=1), this sequence (q=2).

Cf. A000108, A174116, A174117, A174119.

Sequence in context: A157170 A143679 A081581 * A051425 A051469 A155494

Adjacent sequences: A174122 A174123 A174124 * A174126 A174127 A174128

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 09 2010

Extensions

Edited by G. C. Greubel, Feb 11 2021

Status

approved