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 A308654 #40 Aug 15 2020 03:38:44
%S A308654 1,0,2,0,2,2,0,2,4,2,0,2,6,4,2,0,2,8,8,4,2,0,2,10,14,8,4,2,0,2,12,20,
%T A308654 16,8,4,2,0,2,14,28,26,16,8,4,2,0,2,16,38,40,28,16,8,4,2,0,2,18,48,60,
%U A308654 46,28,16,8,4,2,0,2,20,60,84,72,48,28,16,8,4,2
%N A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n.
%C A308654 T(n,0) = A000007(n).
%C A308654 T(n,1) = A040000(n) for n > 0.
%C A308654 T(n,2) = A005843(n-1).
%C A308654 T(n,3) = 2*A007980(n-3).
%C A308654 T(n,4) = 2*A061866(n-1).
%C A308654 T(n,5) = 2*A091773(n-5).
%C A308654 Conjecture: T(n,k) = 2*(the associated Poincaré series). If T(n,1) were 1 for n>0, then T(n, k>1) would be a Poincaré series.
%F A308654 Sum_{k=0..n} T(n,k) = A015128(n), the number of overpartitions of n.
%F A308654 If k > n, T(n,k) = 0.
%F A308654 If n >= k > n/2, T(n,k) = 2*A015128(n-k).
%F A308654 Conjecture: T(n,k) = T(n-k, k) + 2*(T(n-k, k-1) + ... + T(n-k, 1)).
%F A308654 Conjecture: T(n,k) = T(n-1, k-1) + T(n-k, k-1) + T(n-k-1, k-1) + T(n-2k, k-1) + T(n-2k-1) + ...
%F A308654 Conjecture: T(n,1) + T(n-1,2) + ... + T(n-floor(n/2),floor(n/2)) = A300415(n+1).
%F A308654 T(n,2) = 2n - 2.
%F A308654 Conjecture: g.f. T(n,k) = 2*(1+x)(1+x^2)...(1+x^(k-1))/((1-x)...(1-x^k)).
%F A308654 Sum_{k=1..n} k * T(n,k) = A235792(n). - _Alois P. Heinz_, Jun 15 2019
%e A308654 T(5,3) = 8 and counts the overpartitions 3,1,1; 3',1,1; 3,1',1; 3',1',1; 2,2,1; 2',2,1; 2,2,1' and 2',2,1'.
%e A308654 T(16,5) = 404 = T(11,5) + 2*( T(11,4) + T(11,3) + T(11,2) + T(11,1)) = 72 + 2*(84 + 60 + 20 + 2) = 404.
%e A308654 T(16, 5) = T(15,4) + T(11,4) + T(10,4) + T(6,4) + T(5,4) = 248 + 84 + 60 + 8 + 4 = 404.
%e A308654 T(9,1) + T(8,2) + T(7,3) + T(6,4) + T(6,5)= 2 + 14 + 20 + 8 + 2 = 46 =A300415(10).
%e A308654 Triangle: T(n,k) begins:
%e A308654 1;
%e A308654 0, 2;
%e A308654 0, 2, 2;
%e A308654 0, 2, 4, 2;
%e A308654 0, 2, 6, 4, 2;
%e A308654 0, 2, 8, 8, 4, 2;
%e A308654 0, 2, 10, 14, 8, 4, 2;
%e A308654 0, 2, 12, 20, 16, 8, 4, 2;
%e A308654 0, 2, 14, 28, 26, 16, 8, 4, 2;
%e A308654 0, 2, 16, 38, 40, 28, 16, 8, 4, 2;
%e A308654 0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2;
%e A308654 ...
%p A308654 b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A308654 expand(`if`(j>0, 2*x^j, 1)*b(n-i*j, i-1)), j=0..n/i)))
%p A308654 end:
%p A308654 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A308654 seq(T(n), n=0..14); # _Alois P. Heinz_, Jun 15 2019
%t A308654 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[If[j > 0, 2*x^j, 1]*b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A308654 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
%t A308654 T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, Dec 10 2019, after _Alois P. Heinz_ *)
%Y A308654 Row sums give A015128.
%Y A308654 Main diagonal T(n,n) gives A040000.
%Y A308654 Cf. A005843, A007980, A061866, A091773, A235792, A300415.
%K A308654 nonn,tabl
%O A308654 0,3
%A A308654 _Gregory L. Simay_, Jun 14 2019