%I #15 Aug 29 2024 02:39:33
%S 1,3,4,6,10,9,10,18,21,16,15,28,36,36,25,21,40,54,60,55,36,28,54,75,
%T 88,90,78,49,36,70,99,120,130,126,105,64,45,88,126,156,175,180,168,
%U 136,81,55,108,156,196,225,240,238,216,171,100,66,130,189,240,280,306,315,304
%N A002260 squared, as an infinite lower triangular matrix, read by rows.
%C A 4-dimensional pyramidal triangle.
%C Sum of terms in n-th row = A001296(n-1), 4-dimensional pyramidal numbers. A001296 = 1, 6, 25, 65, 140. ... E.g.: sum of terms in 5th row of A095729 = (15+28+36+36+25) = 140 = A001296(4). 2. By columns, k; even columns sequences as f(x), x = 1, 2, 3...; = (k/2)x^2 + (k^2 - k/2)x. For example, terms in row 2, (A028552): 4, 10, 18, 28, 40...= x^2 + 3x; row 4 = 2x^2 + 14x, row 6 = 3x^2 + 33x, row 8 = 4x^2 + 60x...etc.
%F Square of an n X n matrix of the form (exemplified by n=3) [1 0 0 / 1 2 0 / 1 2 3]; generates the first n rows of the triangle; where each n-th row starting with 1, has n terms: 1; 3, 4; 6, 10, 9; 10, 18, 21, 16;...
%F The number in the i-th row and j-th column (j<=i) of the squared matrix is j*(binomial[i + 1, 2] - binomial[j, 2]). - Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007
%e First few rows of the triangle are
%e 1;
%e 3, 4;
%e 6, 10, 9;
%e 10, 18, 21, 16;
%e 15, 28, 36, 36, 25;
%e 21, 40, 54, 60, 55, 36,
%e ...
%e [1 0 0 / 1 2 0 / 1 2 3]^2 = [1 0 0 / 3 4 0 / 6 10 9]. Next higher order matrix generates rows of the one lower order, plus the next row: For example, the 4 X 4 matrix [1 0 0 0 / 1 2 0 0 / 1 2 3 0 / 1 2 3 4]^2 = [1 0 0 0 / 3 4 0 0 / 6 10 9 0 / 10 18 21 16].
%t FindRow[n_] := Module[{i = 0}, While[Binomial[i, 2] < n, i++ ]; i - 1]; FindCol[n_] := n - Binomial[FindRow[n], 2]; A095729[n_] := FindCol[n](Binomial[FindRow[n]+1, 2] - Binomial[FindCol[n], 2]); Table[A095729[i], {i, 1, 91}] - Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007
%Y Cf. A001296, A028552, A002260.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_, Jun 05 2004, Feb 17 2007
%E More terms from Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007
%E Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_