cp's OEIS Frontend

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

%I A124326 #37 Dec 29 2024 10:56:27
%S A124326 1,3,3,6,10,6,10,22,22,10,15,40,53,40,15,21,65,105,105,65,21,28,98,
%T A124326 185,226,185,98,28,36,140,301,431,431,301,140,36,45,192,462,756,887,
%U A124326 756,462,192,45,55,255,678,1246,1673,1673,1246,678,255,55,66,330,960,1956,2954
%N A124326 T(n,m) = A007318(n,m) - A077028(n,m).
%C A124326 First term of n-th row is n*(n+1)/2.
%C A124326 Row sum are A002663 (without initial zeros).
%C A124326 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.
%C A124326 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
%H A124326 Iva Kodrnja and Helena Koncul, <a href="https://arxiv.org/abs/2405.10747">Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N))</a>, arXiv:2405.10747 [math.NT], 2024. See p. 10, also <a href="https://hrcak.srce.hr/file/470068">Glasnik Matematički</a>, (2024) Vol. 59, No. 79, 313-325. See pp. 319-320.
%F A124326 T(n,m) = A007318(n,m) - A077028(n,m) (skipping zeros).
%e A124326 Table begins
%e A124326    1;
%e A124326    3,   3;
%e A124326    6,  10,   6;
%e A124326   10,  22,  22,  10;
%e A124326   15,  40,  53,  40,  15;
%e A124326   21,  65, 105, 105,  65,  21;
%e A124326   28,  98, 185, 226, 185,  98,  28;
%e A124326   36, 140, 301, 431, 431, 301, 140,  36;
%e A124326   45, 192, 462, 756, 887, 756, 462, 192,  45;
%e A124326   ...
%e A124326 If the zeros are included, the table begins
%e A124326    0;
%e A124326    0, 0;
%e A124326    0, 0, 0;
%e A124326    0, 0, 0, 0;
%e A124326    0, 0, 1, 0, 0;
%e A124326    0, 0, 3, 3, 0, 0;
%e A124326    0, 0, 6, 10, 6, 0, 0;
%e A124326    0, 0, 10, 22, 22, 10, 0, 0;
%e A124326    ... - from _Michael De Vlieger_, May 27 2024
%t A124326 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}]
%Y A124326 Cf. A077028, A007318, A002663, A014430.
%K A124326 nonn,tabf,uned
%O A124326 1,2
%A A124326 _Roger L. Bagula_, Jun 26 2007