A124326 T(n,m) = A007318(n,m) - A077028(n,m).

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

Author

Roger L. Bagula, Jun 26 2007

Keywords

• nonn
• tabf
• uned

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

Examples

			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
		

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, also Glasnik Matematički, (2024) Vol. 59, No. 79, 313-325. See pp. 319-320.

Crossrefs

Cf. A077028, A007318, A002663, A014430.

Programs

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}]

Formula

T(n,m) = A007318(n,m) - A077028(n,m) (skipping zeros).