This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124840 #16 Jun 01 2025 03:45:34
%S A124840 1,1,-2,1,-4,2,1,-6,6,-1,1,-8,12,-4,-2,1,-10,20,-10,-10,10,1,-12,30,
%T A124840 -20,-30,60,-30,1,-14,42,-35,-70,210,-210,76,1,-16,56,-56,-140,560,
%U A124840 -840,608,-173,1,-18,72,-84,-252,1260,-2520,2736,-1557,363,1,-20,90,-120,-420,2520,-6300,9120,-7785,3630,-717
%N A124840 Triangle, row sums = A008683, the Mobius sequence.
%C A124840 Cf. A124839, the inverse binomial transform of mu(n), A008683.
%F A124840 Binomial transform of the diagonalized matrix using A124839; i.e., let A124839 (1, -2, 2, -1...) = the diagonal of an infinite matrix M; then the triangle (deleting the zeros) = P*M where P = Pascal's triangle as an infinite lower triangular matrix.
%e A124840 First few rows of the triangle are:
%e A124840 1;
%e A124840 1, -2;
%e A124840 1, -4, 2;
%e A124840 1, -6, 6, -1;
%e A124840 1, -8, 12, -4, -2
%e A124840 1, -10, 20, -10, -10, 10;
%e A124840 1, -12, 30, -20, -30, 60, -30;
%e A124840 ...
%e A124840 E.g., mu(5) = -1 = sum of row 5 terms: (1 - 8 + 12 - 4 - 2).
%t A124840 s[n_] := Sum[(-1)^(n-k) * Binomial[n-1, k-1] * MoebiusMu[k], {k, 1, n}]; row[n_] := Table[s[k+1] * Binomial[n, k], {k, 0, n}]; Array[row, 11, 0] // Flatten (* _Amiram Eldar_, Jun 01 2025 *)
%Y A124840 Cf. A008683, A124839.
%K A124840 sign,tabl
%O A124840 1,3
%A A124840 _Gary W. Adamson_, Nov 10 2006
%E A124840 a(39) corrected and more terms added by _Amiram Eldar_, Jun 01 2025