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

0,8

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

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A000108, A037966.

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

Sequence in context: A375863 A108428 A373431 * A075613 A221837 A189150

Adjacent sequences: A174123 A174124 A174125 * A174127 A174128 A174129

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 09 2010

Extensions

Edited by G. C. Greubel, Feb 10 2021

Status

approved