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 A373547 #21 Jun 12 2024 17:20:24
%S A373547 1,1,4,1,12,16,1,24,80,64,1,40,240,448,256,1,60,560,1792,2304,1024,1,
%T A373547 84,1120,5376,11520,11264,4096,1,112,2016,13440,42240,67584,53248,
%U A373547 16384,1,144,3360,29568,126720,292864,372736,245760,65536,1,180,5280,59136,329472,1025024,1863680,1966080,1114112,262144
%N A373547 Triangle read by rows: T(n,k) = 4^k*binomial(n+k, n-k) with 0 <= k <= n.
%C A373547 T(n,k) is the number of occurrences of the periodic substring (01)^k in the periodic string (0011)^n (see Proposition 4.3 at page 6 in Fang).
%C A373547 The word (w_1, w_2, ..., w_r)^m is defined as the word obtained by concatenating (w_1, w_2, ..., w_r) m times.
%C A373547 A word w' = (w'_1, w'_2, ..., w'_s) is said be a subword of a given word w = (w_1, w_2, ..., w_r), if there is some set P = {p_1 < ... < p_s} of integers from 1 to r satisfying w_{p_j} = w'_j for all 1 <= j <= s, and we call the set P an occurrence of w' in w (see Preliminaries section at pp. 2-3 in Fang).
%H A373547 Wenjie Fang, <a href="http://www.arxiv.org/abs/2406.02971">Maximal number of subword occurrences in a word</a>, arXiv:2406.02971 [math.CO], 2024.
%F A373547 G.f.: (1 - x)/((1 - x)^2 - 4*x*y).
%F A373547 T(n,k) = A000302(k)*A085478(n,k).
%F A373547 Sum_{k=0..n} T(n-k,k) = A046717(n).
%F A373547 T(n,2) = A130810(n+2).
%F A373547 T(n,3) = A130812(n+3).
%e A373547 The triangle begins as:
%e A373547 1;
%e A373547 1, 4;
%e A373547 1, 12, 16;
%e A373547 1, 24, 80, 64;
%e A373547 1, 40, 240, 448, 256;
%e A373547 1, 60, 560, 1792, 2304, 1024;
%e A373547 1, 84, 1120, 5376, 11520, 11264, 4096;
%e A373547 ...
%e A373547 T(2,1) = 12 since there are 12 occurrences of (01)^1 = 01 in (0011)^2 = 00110011: {1, 3}, {1, 4}, {1, 7}, {1, 8}, {2, 3}, {2, 4}, {2, 7}, {2, 8}, {5, 7}, {5, 8}, {6, 7}, {6, 8}.
%t A373547 T[n_,k_]:=4^k Binomial[n+k,n-k]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten (* or *)
%t A373547 T[n_,k_]:=SeriesCoefficient[(1-x)/((1-x)^2-4x y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten
%Y A373547 Cf. A000012 (k=0), A000302 (diagonal), A001653 (row sums), A046092 (k=1), A046717, A085478, A130810, A130812, A373628.
%K A373547 nonn,tabl
%O A373547 0,3
%A A373547 _Stefano Spezia_, Jun 09 2024