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

0,5

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

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.

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)

Example

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;

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 *)

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

Crossrefs

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

Cf. A000108, A174116, A174117, A174119.

Sequence in context: A146772 A202868 A202877 * A174345 A174449 A174150

Adjacent sequences: A174121 A174122 A174123 * A174125 A174126 A174127

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 09 2010

Extensions

Edited by G. C. Greubel, Feb 11 2021

Status

approved