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A144680
Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).
2
1, 2, 3, 3, 5, 7, 4, 7, 11, 14, 5, 9, 15, 21, 25, 6, 11, 19, 28, 36, 41, 7, 13, 23, 35, 47, 57, 63, 8, 15, 27, 42, 58, 73, 85, 92, 9, 17, 31, 49, 69, 89, 107, 121, 129, 10, 19, 35, 56, 80, 105, 129, 150, 166, 175
OFFSET
1,2
COMMENTS
Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).
FORMULA
Sum_{k=1..n} T(n, k) = A006008(n).
From G. C. Greubel, Oct 18 2021: (Start)
T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).
T(n, n) = A004006(n).
T(n, n-1) = A050407(n+2).
T(n, n-2) = A027965(n-1) = A074742(n-2). (End)
EXAMPLE
The array is formed by A004736 * A144328 (transform) where A004736 = the natural number decrescendo triangle and A144328 = a crescendo triangle. First few rows of the array =
1, 1, 1, 1, 1, 1, ...
2, 3, 3, 3, 3, 3, ...
3, 5, 7, 7, 7, 7, ...
4, 7, 11, 14, 14, 14, ...
5, 9, 15, 21, 25, 25, ...
...
Triangle begins as:
1;
2, 3;
3, 5, 7;
4, 7, 11, 14;
5, 9, 15, 21, 25;
6, 11, 19, 28, 36, 41;
7, 13, 23, 35, 47, 57, 63;
8, 15, 27, 42, 58, 73, 85, 92;
9, 17, 31, 49, 69, 89, 107, 121, 129;
10, 19, 35, 56, 80, 105, 129, 150, 166, 175;
...
MATHEMATICA
T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 18 2021 *)
PROG
(SageMath)
def A144680(n, k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6
flatten([[A144680(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 18 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Sep 19 2008
STATUS
approved