cp's OEIS Frontend

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.

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)).

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%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