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 A383735 #8 May 14 2025 13:27:47 %S A383735 1,0,1,0,2,1,0,2,0,1,0,2,0,2,1,0,2,0,2,4,1,0,2,0,2,12,6,1,0,2,0,2,24, %T A383735 18,0,1,0,2,0,2,52,48,0,4,1,0,2,0,2,108,126,0,12,4,1,0,2,0,2,224,300, %U A383735 0,24,12,8,1,0,2,0,2,412,762,0,52,24,32,0,1 %N A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids. %C A383735 T(n,k) is the coefficient of p^(k+1), k >= 0, in the power series expansion of the expected finite size of the cluster containing a given cell for percolation with probability p on the polyominoid cells corresponding to row n of A366766. If the given cell is not open, its cluster is empty. Equivalently, T(n,k) can be taken to be the coefficient of p^k if we condition on the event that the given cell is open. %C A383735 See A366766 for details on how the polyominoids are specified and on the ordering of the rows. %H A383735 <a href="/index/Cl#cluster">Index entries for sequences related to cluster series</a>. %F A383735 T(n,k) = [p^k] Sum_P m^2*p^(m-1)*(1-p)^j / binomial(D,d) = Sum_P m^2*(-1)^(k-m+1)*binomial(j,k-m+1) / binomial(D,d), where the sum is over all fixed polyominoids P (corresponding to row n of A366766), m is the number of cells of P, and j is the number of cells that are not in P but are adjacent to a cell in P; d is the dimension of the cells and D is the dimension of the ambient space. It is sufficient to take the sums over those P that have at most k+1 cells. %e A383735 Array begins: %e A383735 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A383735 ---+----------------------------------------------------------------- %e A383735 1 | 1 0 0 0 0 0 0 0 0 0 0 0 0 %e A383735 2 | 1 2 2 2 2 2 2 2 2 2 2 2 2 %e A383735 3 | 1 0 0 0 0 0 0 0 0 0 0 0 0 %e A383735 4 | 1 2 2 2 2 2 2 2 2 2 2 2 2 %e A383735 5 | 1 4 12 24 52 108 224 412 844 1528 3152 5036 11984 %e A383735 6 | 1 6 18 48 126 300 762 1668 4216 8668 21988 43058 110832 %e A383735 7 | 1 0 0 0 0 0 0 0 0 0 0 0 0 %e A383735 8 | 1 4 12 24 52 108 224 412 844 1528 3152 5036 11984 %e A383735 9 | 1 4 12 24 52 108 224 412 844 1528 3152 5036 11984 %e A383735 10 | 1 8 32 108 348 1068 3180 9216 26452 73708 206872 563200 1555460 %e A383735 11 | 1 0 0 0 0 0 0 0 0 0 0 0 0 %e A383735 12 | 1 2 2 2 2 2 2 2 2 2 2 2 2 %Y A383735 Cf. A366766, A366767, A366768. %Y A383735 Rows include: %Y A383735 n | sequence for row n %Y A383735 ---+------------------- %Y A383735 1 | A000007 %Y A383735 2 | A040000 %Y A383735 3 | A000007 %Y A383735 4 | A040000 %Y A383735 5 | A003203 %Y A383735 6 | A003198 %Y A383735 7 | A000007 %Y A383735 8 | A003203 %Y A383735 9 | A003203 %Y A383735 10 | A003201 %Y A383735 11 | A000007 %Y A383735 12 | A040000 %Y A383735 13 | A383737 %Y A383735 14 | A003207 %Y A383735 15 | A000007 %Y A383735 16 | A003203 %Y A383735 17 | A383737 %Y A383735 18 | A383736 %Y A383735 19 | A003203 %Y A383735 20 | A003201 %Y A383735 ... %Y A383735 31 | A000007 %Y A383735 32 | A003211 %Y A383735 33 | A003209 %Y A383735 34 | A036396 %Y A383735 35 | A003210 %Y A383735 ... %Y A383735 38 | A036402 %Y A383735 39 | A000007 %Y A383735 40 | A040000 %Y A383735 ... %Y A383735 43 | A000007 %Y A383735 44 | A003203 %Y A383735 ... %Y A383735 47 | A003203 %Y A383735 48 | A003201 %K A383735 nonn,tabl %O A383735 1,5 %A A383735 _Pontus von Brömssen_, May 10 2025