A174127 Triangle T(n, k) = (n-k)^3 * binomial(n-1, k-1)^3 with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 1, 1, 1, 8, 8, 1, 1, 27, 216, 27, 1, 1, 64, 1728, 1728, 64, 1, 1, 125, 8000, 27000, 8000, 125, 1, 1, 216, 27000, 216000, 216000, 27000, 216, 1, 1, 343, 74088, 1157625, 2744000, 1157625, 74088, 343, 1, 1, 512, 175616, 4741632, 21952000, 21952000, 4741632, 175616, 512, 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 11 2021

Links

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

Formula

Let c(n) = Product_{i=2..n} (i-1)^3 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 11 2021: (Start)

T(n, k) = (n-k)^3 * binomial(n-1, k-1)^3 with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = 2 + (n-1)^3*A000172(n-2) - [n=0]. (End)

Example

Triangle begins as:
  1;
  1,   1;
  1,   1,      1;
  1,   8,      8,       1;
  1,  27,    216,      27,        1;
  1,  64,   1728,    1728,       64,        1;
  1, 125,   8000,   27000,     8000,      125,       1;
  1, 216,  27000,  216000,   216000,    27000,     216,      1;
  1, 343,  74088, 1157625,  2744000,  1157625,   74088,    343,   1;
  1, 512, 175616, 4741632, 21952000, 21952000, 4741632, 175616, 512, 1;

Mathematica

(* First program *)
c[n_]:= If[n<2, 1, Product[(i-1)^3, {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, 3], {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, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 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, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

Crossrefs

Cf. A155865 (q=1), A174126 (q=2), this sequence (q=3).

Cf. A000172.

Sequence in context: A172352 A141134 A176155 * A230153 A091648 A135707

Adjacent sequences: A174124 A174125 A174126 * A174128 A174129 A174130

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 09 2010

Extensions

Edited by G. C. Greubel, Feb 11 2021

Status

approved