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 A378783 #19 Dec 21 2024 00:49:45 %S A378783 -1,2,1,-5,-1,-2,14,1,5,4,-42,-1,-12,-8,-9,132,1,29,18,22,21,-429,-1, %T A378783 -73,-43,-54,-50,-51,1430,1,190,105,135,124,128,127,-4862,-1,-505, %U A378783 -262,-345,-315,-326,-322,-323,16796,1,1363,666,896,813,843,832,836,835 %N A378783 Triangular array T(n,k) read by rows: T(n, k) = c_k(n+1). The sequence c_k(m) has the ordinary generating function C_k(x) which satisfies C_k(x) = 1/(1+C_k(x)*Sum_{t=0..k} x^(t+1)). %F A378783 G.f. column k: (2 / (sqrt(1+4*Sum_{t=0..k}x^(t+1)) + 1) - 1)/x. %F A378783 T(n, 0) = (-1)^(n+1)*Catalan(n+1) = A168491(n+1). %F A378783 T(n, 2) = (-1)^(n+1)*A152171(n+1). %F A378783 T(n, n) = (-1)^(n+1)*A001006(n) = -A166587(n+1). %F A378783 A378816(n) = Limit_{k->oo} (T(k, k-n) - T(k, k-n-1)). %e A378783 Triangle begins: %e A378783 [0] -1 %e A378783 [1] 2, 1 %e A378783 [2] -5, -1, -2 %e A378783 [3] 14, 1, 5, 4 %e A378783 [4] -42, -1, -12, -8, -9 %e A378783 [5] 132, 1, 29, 18, 22, 21 %e A378783 [6] -429, -1, -73, -43, -54, -50, -51 %e A378783 [7] 1430, 1, 190, 105, 135, 124, 128, 127 %e A378783 [8] -4862, -1, -505, -262, -345, -315, -326, -322, -323 %e A378783 . %t A378783 T[n_,k_]:=SeriesCoefficient[(2 / (Sqrt[1+4*Sum[x^(t+1),{t,0,k}]] + 1) - 1)/x,{x,0,n}];Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* _Stefano Spezia_, Dec 08 2024 *) %o A378783 (PARI) %o A378783 column(n, max_n) = { my(s = 1,x = 'x,cu); for(k = 0, max_n-1, cu = cu+polcoeff(1/s+O(x^(k+1)), k, x); cu = cu-polcoeff(1/s+O(x^(k+1)), k-1-n, x); s = s+cu*x^(k+1)); Vec(1/s+O(x^max_n)) }; %o A378783 T(n, k) = column(k, n+2)[n+2] %o A378783 T(n, k) = polcoeff(2 / (sqrt(1+4*x*sum(t=0, k, x^t)) + 1) + O(x^(n+2)), n+1, x) %Y A378783 Cf. A001006, A000108, A152171, A166587, A168491, A378816. %K A378783 sign,tabl %O A378783 0,2 %A A378783 _Thomas Scheuerle_, Dec 07 2024