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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. 2
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 (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 10 2021
LINKS
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. A155865 (q=1), this sequence (q=2), A174127 (q=3).
Sequence in context: A198929 A172347 A108428 * A075613 A221837 A189150
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Mar 09 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 10 2021
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)