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 A145006 #9 Jun 02 2025 00:37:15
%S A145006 1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,-1,0,0,1,1,0,0,-1,0,0,1,1,0,-1,0,-1,0,
%T A145006 0,1,1,0,0,-1,0,-1,0,0,1,1,0,0,0,-1,0,-1,0,0,1,1,0,0,0,0,-1,0,-1,0,0,
%U A145006 1,1,0,0,0,0,0,-1,0,-1,0,0,1,1,0
%N A145006 Triangle read by rows, generator for the partition numbers, A000041.
%C A145006 The partition numbers, A000041, = eigenvector of the triangle. With A080995, characteristic function of the generalized pentagonal numbers, we apply signs: (++ -- ++,...) to the 1's, starting with offset 1. This gives an opposite parity to Euler's partition formula which is (with offset 1): -p(n-1) - p(n-2) + p(n-5) + p(n-7),...
%C A145006 By applying termwise products of A000041 terms and row terms of A145006, we obtain the eigentriangle of the partition numbers.
%F A145006 Triangle by columns: let A = an infinite lower triangular matrix with the characteristic function of A001318: (1, 2, 5, 7, 12, 15,...) in every column; signed: (++ -- ++,...).
%F A145006 Shift triangle A down one place and insert "1" in the T(0,0) position, giving triangle A145006. The eigenvector of the triangle = A000041, the partition numbers: (1, 1, 2, 3, 5, 7, 11,...). Lim_{n=1..inf} A145006^n = A000041. Or, simply take a suitably large power of the triangle, which quickly converges to A000041 as a vector.
%e A145006 First few rows of the triangle =
%e A145006 1;
%e A145006 1, 0;
%e A145006 1, 1, 0;
%e A145006 0, 1, 1, 0;
%e A145006 0, 0, 1, 1, 0;
%e A145006 -1, 0, 0, 1, 1, 0;
%e A145006 0, -1, 0, 0, 1, 1, 0;
%e A145006 -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 ...
%Y A145006 A000041, Cf. A080995, A001318, A145007
%K A145006 eigen,tabl,sign
%O A145006 0,1
%A A145006 _Gary W. Adamson_, Sep 28 2008