OFFSET
0,14
COMMENTS
Rows, of the square array, are three-fold convolutions of sequences of powers.
LINKS
G. C. Greubel, Antidiadoganal rows n = 0..50, flattened
FORMULA
T(n, k) = k^(n-2)*binomial(n, 2), with T(n, 0) = 0 (square array).
T(n, n) = A081131(n).
Rows have g.f. x^3/(1-k*x)^n.
From G. C. Greubel, May 14 2021: (Start)
T(k, n-k) = (n-k)^(k-2)*binomial(k,2) with T(n, n) = 0 (antidiagonal triangle).
Sum_{k=0..n} T(n, n-k) = A081197(n). (End)
EXAMPLE
The array begins as:
0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, ...
0, 1, 1, 1, 1, 1, ... A000012
0, 3, 6, 9, 12, 15, ... A008585
0, 6, 24, 54, 96, 150, ... A033581
0, 10, 80, 270, 640, 1250, ... A244729
The antidiagonal triangle begins as:
0;
0, 0;
0, 0, 0;
0, 0, 1, 0;
0, 0, 1, 3, 0;
0, 0, 1, 6, 6, 0;
0, 0, 1, 9, 24, 10, 0;
MATHEMATICA
Table[If[k==n, 0, (n-k)^(k-2)*Binomial[k, 2]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2021 *)
PROG
(Magma) [k eq n select 0 else (n-k)^(k-2)*Binomial(k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, May 14 2021
(Sage) flatten([[0 if (k==n) else (n-k)^(k-2)*binomial(k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 14 2021
(PARI)
T(n, k)=if (k==0, 0, k^(n-2)*binomial(n, 2));
seq(nn) = for (n=0, nn, for (k=0, n, print1(T(k, n-k), ", ")); );
seq(12) \\ Michel Marcus, May 14 2021
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 08 2003
EXTENSIONS
Term a(5) corrected by G. C. Greubel, May 14 2021
STATUS
approved