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A095729 A002260 squared, as an infinite lower triangular matrix, read by rows.

%I A095729 #24 Aug 13 2025 10:19:21
%S A095729 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 A095729 88,90,78,49,36,70,99,120,130,126,105,64,45,88,126,156,175,180,168,
%U A095729 136,81,55,108,156,196,225,240,238,216,171,100,66,130,189,240,280,306,315,304
%N A095729 A002260 squared, as an infinite lower triangular matrix, read by rows.
%C A095729 Sum of terms in n-th row = A001296(n-1).
%C A095729 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.
%C A095729 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 A095729 First few rows of the triangle are
%e A095729   1;
%e A095729   3, 4;
%e A095729   6, 10, 9;
%e A095729   10, 18, 21, 16;
%e A095729   15, 28, 36, 36, 25;
%e A095729   21, 40, 54, 60, 55, 36,
%e A095729   ...
%e A095729 [1 0 0 / 1 2 0 / 1 2 3]^2 = [1 0 0 / 3 4 0 / 6 10 9].
%e A095729 Next higher order matrix generates rows of the one lower order, plus the next row.
%e A095729 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 A095729 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 A095729 Cf. A001296, A028552, A002260.
%K A095729 nonn,tabl
%O A095729 1,2
%A A095729 _Gary W. Adamson_, Jun 05 2004, Feb 17 2007
%E A095729 More terms from Keith Schneider (schneidk(AT)email.unc.edu), Jul 23 2007
%E A095729 Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_