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 A374839 #16 Aug 24 2024 13:53:20
%S A374839 1,3,1,7,7,1,15,35,23,1,31,155,403,279,1,63,651,6603,71827,65815,1,
%T A374839 127,2667,106299,18394315,4313323667,4295033111,1,255,10795,1703451,
%U A374839 4709050939,282677998234827,18447026751295461523,18446744078004584727,1
%N A374839 Triangle read by rows: T(n,k) is the number of strong subtrees of height k in the complete binary tree of height n.
%C A374839 Informally, a strong subtree is one that preserves meets, the relative level of vertices, and the number of immediate successors to non-terminal vertices.
%C A374839 The collection of strong subtrees with height k of some tree T is often denoted by S_k(T). T(n,k) is the cardinality of S_k(2^(<n)).
%C A374839 There is a general expression for T(n,k) given in terms of an auxiliary function that can be eliminated for the diagonals, with the following examples:
%C A374839 T(m+1,m) = 3 + Sum_{j=1..m-1} 2^(2^i).
%C A374839 T(m+2,m) = 7 + 3*(Sum_{j=1..m-1} 2^(2^i)) + Sum_{1<=i,j<=m-1} 2^(2^i + 2^j).
%C A374839 T(m+3,m) = 15 + 7*(Sum_{j=1..m-1} 2^(2^i)) + 3*(Sum_{1<=i,j<=m-1} 2^(2^i + 2^j)) + Sum_{1<=i,j,k<=m-1} 2^(2^i + 2^j + 2^k).
%H A374839 John V. Siratt, <a href="https://curate.nd.edu/articles/dataset/Some_Applications_of_Formal_Mathematics/26195681">Some applications of formal mathematics</a>, Doctoral dissertation, University of Notre Dame (2024).
%F A374839 T(n,k) = Sum_{1 <= x_1 < ... < x_k <= n} Product_{i=1..k} (2^(x_i - x_{i-1} - 1))^(2^(i - 1)), where x_0 = 0.
%e A374839 Triangle begins:
%e A374839 1;
%e A374839 3, 1;
%e A374839 7, 7, 1;
%e A374839 15, 35, 23, 1;
%e A374839 31, 155, 403, 279, 1;
%e A374839 63, 651, 6603, 71827, 65815, 1;
%e A374839 127, 2667, 106299, 18394315, 4313323667, 4295033111, 1;
%e A374839 ...
%e A374839 Formatted as a transposed array:
%e A374839 T(n,k) | n=1 2 3 4 5 6 7 8
%e A374839 --------------------------------------------------------------------
%e A374839 k=1 | 1 3 7 15 31 63 127 255
%e A374839 2 | 0 1 7 35 155 651 2667 10795
%e A374839 3 | 0 0 1 23 403 6603 106299 1703451
%e A374839 4 | 0 0 0 1 279 71827 18394315 4709050939
%e A374839 5 | 0 0 0 0 1 65815 4313323667 282677998234827
%e A374839 6 | 0 0 0 0 0 1 4295033111 18447026751295461523
%e A374839 7 | 0 0 0 0 0 0 1 18446744078004584727
%e A374839 8 | 0 0 0 0 0 0 0 1
%o A374839 (PARI) T(n,k)={my(s=0); forvec(x=vector(k,i,[1,n]),s+=prod(i=1, k, (2^(x[i] - if(i>1, x[i-1]) - 1))^(2^(i - 1))), 2); s}
%o A374839 { for(n=1, 8, print(vector(n,k,T(n,k)))) } \\ _Andrew Howroyd_, Jul 23 2024
%Y A374839 Column 1 is A000225.
%Y A374839 Column 2 appears to be A006095.
%K A374839 nonn,tabl
%O A374839 1,2
%A A374839 _John V Siratt_, Jul 21 2024