|
|
A287326
|
|
Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.
|
|
15
|
|
|
1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 25, 19, 1, 1, 25, 37, 37, 25, 1, 1, 31, 49, 55, 49, 31, 1, 1, 37, 61, 73, 73, 61, 37, 1, 1, 43, 73, 91, 97, 91, 73, 43, 1, 1, 49, 85, 109, 121, 121, 109, 85, 49, 1, 1, 55, 97, 127, 145, 151, 145, 127, 97, 55, 1, 1, 61, 109, 145, 169, 181, 181, 169, 145, 109, 61, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(1, n, k), i.e T(n, k) is partial case of L(m, n, k) for m = 1.
T(n, k) is symmetric: T(n, k) = T(n, n-k).
(End)
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = 6*k*(n-k) + 1.
G.f. of column k: n^k*(1+(6*k-1)*n)/(1-n)^2.
G.f.: (-1 + 8*y + 5*y^2 + x*(1 - 14*y + y^2))/((-1 + x)^2*(-1 + y)^3). - Stefano Spezia, Oct 09 2018
T(n, k) = 1/2 * T(A294317(n, k), k) + 1/2.
T(n+1, k) = 2*T(n, k) - T(n-1, k), for n >= k.
Sum_{k=0..n-1} T(n, k) = Sum_{k=1..n} T(n, k) = A000578(n).
Sum_{k=1..n-1} T(n, k) = A068601(n).
(n+1)^3 - n^3 = T(A000124(n), 1). (End)
|
|
EXAMPLE
|
Triangle begins:
----------------------------------------
k= 0 1 2 3 4 5 6 7 8
----------------------------------------
n=0: 1;
n=1: 1, 1;
n=2: 1, 7, 1;
n=3: 1, 13, 13, 1;
n=4: 1, 19, 25, 19, 1;
n=5: 1, 25, 37, 37, 25, 1;
n=6: 1, 31, 49, 55, 49, 31, 1;
n=7: 1, 37, 61, 73, 73, 61, 37, 1;
n=8: 1, 43, 73, 91, 97, 91, 73, 43, 1;
|
|
MAPLE
|
T := (n, k) -> 6*k*(n-k) + 1:
|
|
MATHEMATICA
|
T[n_, k_] := 6 k (n - k) + 1; Column[Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jun 02 2019 *)
|
|
PROG
|
(PARI) t(n, k) = 6*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */
(GAP) Flat(List([0..11], n->List([0..n], k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018
(Magma) /* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018
|
|
CROSSREFS
|
Various cases of L(m, n, k): This sequence (m=1), A300656(m=2), A300785(m=3). See comments for L(m, n, k).
Differences of cubes n^3 are T(A000124(n), 1).
Cf. A000578, A038593, A294317, A007318, A055012, A077028, A008458, A302971, A304042, A068601, A166873, A003154, A227776, A000124, A003215, A094053.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|