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 A186020 #37 Apr 09 2021 09:43:39
%S A186020 1,1,1,2,1,1,5,3,1,1,15,9,4,1,1,52,31,14,5,1,1,203,121,54,20,6,1,1,
%T A186020 877,523,233,85,27,7,1,1,4140,2469,1101,400,125,35,8,1,1,21147,12611,
%U A186020 5625,2046,635,175,44,9,1,1,115975,69161,30846,11226,3488,952,236,54,10,1,1
%N A186020 Eigentriangle of the binomial matrix.
%C A186020 Reversal of Gould triangle A121207. First column is A000110. Second column is A040027.
%C A186020 Row sums are A186021. Diagonal sums are A186022.
%C A186020 Construction is described by Paul D. Hanna in A121207. The method of construction is general for this class of eigentriangle.
%H A186020 G. C. Greubel, <a href="/A186020/b186020.txt">Rows n = 0..100 of triangle, flattened</a>
%H A186020 Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv:1107.5490 [math.CO], 2011.
%H A186020 Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
%F A186020 Lower triangular (infinite) matrix T = (U - D*P)^{-1} with the unit matrix U, the Pascal matrix P from A007318 and the matrix D with elements delta_{i,j+1}, for i, j >= 0 (row 0 has only 0s). From the _Paul Barry_ paper rewritten in matrix notation. T satisfies P*T = D'*(T - U), with D' the transposed matrix D, that is the diagonal of T has been erased and the row index shifted on the r.h.s. (showing that the name Eigentriangle or -matrix is a misnomer). For finite N X N matrices P*T = D'*(T - U), only up to the last row. - _Wolfdieter Lang_, Apr 07 2021
%e A186020 Triangle T begins
%e A186020 1;
%e A186020 1, 1;
%e A186020 2, 1, 1;
%e A186020 5, 3, 1, 1;
%e A186020 15, 9, 4, 1, 1;
%e A186020 52, 31, 14, 5, 1, 1;
%e A186020 203, 121, 54, 20, 6, 1, 1;
%e A186020 877, 523, 233, 85, 27, 7, 1, 1;
%e A186020 4140, 2469, 1101, 400, 125, 35, 8, 1, 1;
%e A186020 21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1;
%e A186020 115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1;
%e A186020 Inverse is the identity matrix I minus binomial matrix B shifted down once, or
%e A186020 T^{-1}(n,k)=if(k=n,1,if(k<n,-binomial(n-1,k),0)). This begins
%e A186020 1;
%e A186020 -1, 1;
%e A186020 -1, -1, 1;
%e A186020 -1, -2, -1, 1;
%e A186020 -1, -3, -3, -1, 1;
%e A186020 -1, -4, -6, -4, -1, 1;
%e A186020 -1, -5, -10, -10, -5, -1, 1;
%e A186020 -1, -6, -15, -20, -15, -6, -1, 1;
%e A186020 -1, -7, -21, -35, -35, -21, -7, -1, 1;
%e A186020 -1, -8, -28, -56, -70, -56, -28, -8, -1, 1;
%e A186020 Production matrix is
%e A186020 1, 1;
%e A186020 1, 0, 1;
%e A186020 2, 1, 0, 1;
%e A186020 5, 3, 1, 0, 1;
%e A186020 15, 9, 4, 1, 0, 1;
%e A186020 52, 31, 14, 5, 1, 0, 1;
%e A186020 203, 121, 54, 20, 6, 1, 0, 1;
%e A186020 877, 523, 233, 85, 27, 7, 1, 0, 1;
%e A186020 4140, 2469, 1101, 400, 125, 35, 8, 1, 0, 1;
%e A186020 21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 0, 1;
%t A186020 t[n_, k_] := t[n, k] = If[k == 0, 1, Sum[t[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]];
%t A186020 T[n_, k_] := t[n, n-k];
%t A186020 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 27 2018 *)
%Y A186020 Cf. A000110, A007318, A121207, A124496, A160185, A186021, A186022.
%K A186020 nonn,easy,tabl
%O A186020 0,4
%A A186020 _Paul Barry_, Feb 10 2011