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A124326
T(n,m) = A007318(n,m) - A077028(n,m).
1
1, 3, 3, 6, 10, 6, 10, 22, 22, 10, 15, 40, 53, 40, 15, 21, 65, 105, 105, 65, 21, 28, 98, 185, 226, 185, 98, 28, 36, 140, 301, 431, 431, 301, 140, 36, 45, 192, 462, 756, 887, 756, 462, 192, 45, 55, 255, 678, 1246, 1673, 1673, 1246, 678, 255, 55, 66, 330, 960, 1956, 2954
OFFSET
1,2
COMMENTS
First term of n-th row is n*(n+1)/2.
Row sum are A002663 (without initial zeros).
Appears to be the triangle resulting from adding the row number (first row numbered 0) of Pascal's triangle (A007318) to each entry in that row, subtracting the corresponding entries in the triangle formed by taking the finite diagonals in the multiplication table in order of increasing length (A003991), and removing the outer two layers, which consist entirely of 0's.
Each value of the sequence T(x,y) is equal to the sum of all values in A014430 that are in the rectangle defined by the tip (0,0) and the position (x,y). - Jon Perry, Sep 11 2013
LINKS
Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
FORMULA
T(n,m) = A007318(n,m) - A077028(n,m) (skipping zeros).
EXAMPLE
Table begins
1;
3, 3;
6, 10, 6;
10, 22, 22, 10;
15, 40, 53, 40, 15;
21, 65, 105, 105, 65, 21;
28, 98, 185, 226, 185, 98, 28;
36, 140, 301, 431, 431, 301, 140, 36;
45, 192, 462, 756, 887, 756, 462, 192, 45;
...
If the zeros are included, the table begins
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 1, 0, 0;
0, 0, 3, 3, 0, 0;
0, 0, 6, 10, 6, 0, 0;
0, 0, 10, 22, 22, 10, 0, 0;
... - from Michael De Vlieger, May 27 2024
MATHEMATICA
a = Table[Flatten[Table[If[Binomial[m, n] - (1 +n (m - n)) == 0, {}, Binomial[m, n] - (1 + n (m - n))], {n, 0, m}]], {m, 0, 14}]
CROSSREFS
KEYWORD
nonn,tabf,uned
AUTHOR
Roger L. Bagula, Jun 26 2007
STATUS
approved