A174126 Triangle T(n, k) = (n-k)^2 * binomial(n-1, k-1)^2 with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 9, 36, 9, 1, 1, 16, 144, 144, 16, 1, 1, 25, 400, 900, 400, 25, 1, 1, 36, 900, 3600, 3600, 900, 36, 1, 1, 49, 1764, 11025, 19600, 11025, 1764, 49, 1, 1, 64, 3136, 28224, 78400, 78400, 28224, 3136, 64, 1, 1, 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81, 1

Offset: 0

Author

Roger L. Bagula, Mar 09 2010

Keywords

• nonn
• tabl
• easy

Comments

This triangle sequence is part of a class of triangles defined by T(n, k, q) = (n-k)^q * binomial(n-1, k-1)^q with T(n, 0) = T(n, n) = 1 and have row sums Sum_{k=0..n} T(n, k, q) = 2 - [n=0] + Sum_{k=1..n-1} k^q * binomial(n-1, k)^q. - G. C. Greubel, Feb 10 2021

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,    1;
  1,  4,    4,     1;
  1,  9,   36,     9,      1;
  1, 16,  144,   144,     16,      1;
  1, 25,  400,   900,    400,     25,      1;
  1, 36,  900,  3600,   3600,    900,     36,     1;
  1, 49, 1764, 11025,  19600,  11025,   1764,    49,    1;
  1, 64, 3136, 28224,  78400,  78400,  28224,  3136,   64,  1;
  1, 81, 5184, 63504, 254016, 396900, 254016, 63504, 5184, 81, 1;
		

Links

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

Crossrefs

Cf. A000108, A037966.

Cf. A155865 (q=1), this sequence (q=2), A174127 (q=3).

Programs

Magma

T:= func< n,k,q | k eq 0 or k eq n select 1 else (n-k)^q*Binomial(n-1,k-1)^q >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

Mathematica

(* First program *)
c[n_]:= If[n<2, 1, Product[(i-1)^2, {i,2,n}]];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_, q_]:= If[k==0 || k==n, 1, (n-k)^q*Binomial[n-1, k-1]^q];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2021 *)

Sage

def T(n,k,q): return 1 if (k==0 or k==n) else (n-k)^q*binomial(n-1,k-1)^q
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021

Formula

Let c(n) = Product_{i=2..n} (i-1)^2 for n > 2 otherwise 1. The number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).

From G. C. Greubel, Feb 10 2021: (Start)

T(n, k) = (n-k)^2 * binomial(n-1, k-1)^2 with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2 + A037966(n-1) - [n=0] = 2 + (n-1)^3*C_{n-2} - [n=0], where C_{n} are the Catalan numbers (A000108) and [] is the Iverson bracket. (End)

Extensions

Edited by G. C. Greubel, Feb 10 2021