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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 13 00:57 EDT 2021. Contains 344980 sequences. (Running on oeis4.)