A174124 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 = 1, read by rows.

1, 1, 1, 1, 6, 1, 1, 12, 12, 1, 1, 20, 40, 20, 1, 1, 30, 100, 100, 30, 1, 1, 42, 210, 350, 210, 42, 1, 1, 56, 392, 980, 980, 392, 56, 1, 1, 72, 672, 2352, 3528, 2352, 672, 72, 1, 1, 90, 1080, 5040, 10584, 10584, 5040, 1080, 90, 1, 1, 110, 1650, 9900, 27720, 38808, 27720, 9900, 1650, 110, 1

Offset: 0

Author

Roger L. Bagula, Mar 09 2010

Keywords

• nonn
• tabl

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

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   6,    1;
  1,  12,   12,    1;
  1,  20,   40,   20,     1;
  1,  30,  100,  100,    30,     1;
  1,  42,  210,  350,   210,    42,     1;
  1,  56,  392,  980,   980,   392,    56,    1;
  1,  72,  672, 2352,  3528,  2352,   672,   72,    1;
  1,  90, 1080, 5040, 10584, 10584,  5040, 1080,   90,   1;
  1, 110, 1650, 9900, 27720, 38808, 27720, 9900, 1650, 110, 1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475
• Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.

Crossrefs

Cf. this sequence (q=1), A174125 (q=2).

Cf. A000108, A174116, A174117, A174119.

Programs

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,1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

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,1], {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,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)

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,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

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 = 1.

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 = 1.

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

Extensions

Edited by G. C. Greubel, Feb 11 2021