OFFSET
0,2
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Triangle read by rows, A000012 * A145677 * A000012; where A000012 = an infinite lower triangular matrix: (1; 1,1; 1,1,1; ...), with all 1's.
From G. C. Greubel, Dec 26 2021: (Start)
T(n, k) = (n+1-k)*(n+k)/2 with T(n, 0) = binomial(n+2, 2).
Sum_{k=0..n} T(n, k) = (1/3)*(n+1)*(n^2 + 2*n + 3) = A006527(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = binomial(n+2, 2) + A034828(n+1).
T(n, 1) = A000217(n).
T(n, 2) = A000096(n-1).
T(n, 3) = A055998(n-2).
T(2*n, n) = A134479(n). (End)
EXAMPLE
First few rows of the triangle =
1;
3, 1;
6, 3, 2;
10, 6, 5, 3;
15, 10, 9, 7, 4;
21, 15, 14, 12, 9, 5;
28, 21, 10, 18, 15, 11, 6;
36, 28, 27, 25, 22, 18, 13, 7;
45, 36, 35, 33, 30, 26, 21, 15, 8;
55, 45, 44, 42, 39, 35, 30, 24, 17, 9;
66, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10;
78, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11;
91, 78, 77, 75, 72, 68, 63, 57, 50, 42, 33, 23, 12;
...
MATHEMATICA
T[n_, k_]:= If[k==0, Binomial[n+2, 2], (n+1-k)*(n+k)/2];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 26 2021 *)
PROG
(Sage)
def A158822(n, k):
if (k==0): return binomial(n+2, 2)
else: return (n-k+1)*(n+k)/2
flatten([[A158822(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 26 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson and Roger L. Bagula, Mar 28 2009
EXTENSIONS
Definition corrected by Michael Somos, Nov 05 2011
STATUS
approved