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 A078121 #48 Sep 01 2025 21:48:49
%S A078121 1,1,1,1,2,1,1,4,4,1,1,10,16,8,1,1,36,84,64,16,1,1,202,656,680,256,32,
%T A078121 1,1,1828,8148,10816,5456,1024,64,1,1,27338,167568,274856,174336,
%U A078121 43680,4096,128,1,1,692004,5866452,11622976,8909648,2794496,349504,16384,256,1
%N A078121 Infinite lower triangular matrix, M, that satisfies [M^2](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.
%C A078121 M also satisfies: [M^(2k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(2^n)](i,j) = M(i+n,j+n) for all n>=0.
%H A078121 Alois P. Heinz, <a href="/A078121/b078121.txt">Rows n = 0..80, flattened</a>
%H A078121 MathOverflow, <a href="https://mathoverflow.net/a/495928/231922">Closed form for diagonals of A078121</a>, (2025).
%F A078121 M(1,j) = A002577(j) (partitions of 2^j into powers of 2), M(j+1,j) = 2^j, M(j+2,j) = 4^j, M(j+3,j) = A016131(j).
%F A078121 M(n,k) = the coefficient of x^(2^n - 2^(n-k)) in the power series expansion of 1/Product_{j=0..n-k} (1-x^(2^j)) whenever 0<=k<n for all n>0 (conjecture).
%F A078121 M(n,k) = Sum_{j=0..n-k-1} M(n-k,j)*M(k+j,k-1)*(1+(-1)^(n+k+j+1)) for 0 < k < n with M(n,0) = M(n,n) = 1. - _Mikhail Kurkov_, Jun 01 2025
%F A078121 From _Mikhail Kurkov_, Jul 01 2025: (Start)
%F A078121 Conjecture 1: let R(n,x) be the n-th row polynomial, then R(n,x) = x*R(n-1,x) + Sum_{k=1..n-1} M(n-1,k-1)*R(k,x)*(-1)^(n+k+1) = R(n-1,x) + x*Sum_{k=1..n-1} (M(n-1,k) - M(n-2,k))*R(k,x) for n > 1 with R(0,x) = 1, R(1,x) = x + 1.
%F A078121 Conjecture 2: M(n+m,n) ~ 2^(m*(2*n+m-1)/2)/m! as n -> oo. More generally, it also looks like that M(n+m,n) for m > 0 can be represented as (Sum_{j=0..flooor((m-1)/2)} 2^((m-2*j)*(2*(n-j)+m-1)/2)*P(m,j)*(-1)^j)/m! where P(m,j) are some positive integer coefficients. (End)
%e A078121 The square of the matrix is the same matrix excluding the first row and column:
%e A078121 [1, 0, 0, 0, 0]^2 = [ 1, 0, 0, 0, 0]
%e A078121 [1, 1, 0, 0, 0] [ 2, 1, 0, 0, 0]
%e A078121 [1, 2, 1, 0, 0] [ 4, 4, 1, 0, 0]
%e A078121 [1, 4, 4, 1, 0] [10,16, 8, 1, 0]
%e A078121 [1,10,16, 8, 1] [36,84,64,16, 1]
%p A078121 M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
%p A078121 add(M(i-1, k)*M(k, j-1), k=0..i-1))
%p A078121 end:
%p A078121 seq(seq(M(n,k), k=0..n), n=0..10); # _Alois P. Heinz_, Feb 27 2015
%t A078121 rows = 10; M[k_] := Table[ Which[j == 1, 1, i == j, 1, 1 < j < i, m[i, j], True, 0], {i, 1, k}, {j, 1, k}]; m2[i_, j_] := m[i+1, j+1]; M2[k_] := Table[ Which[j<i, m2[i, j], j == i, 1, True, 0], {i, 1, k}, {j, 1, k}]; sol[k_] := Thread[ Flatten[ M[k].M[k]] == Flatten[M2[k]]] // Solve; Table[M[rows][[i, j]], {i, 1, rows}, {j, 1, i}] /. sol[rows] // Flatten (* _Jean-François Alcover_, Feb 27 2015 *)
%t A078121 M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[M[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[M[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 27 2015, after _Alois P. Heinz_ *)
%o A078121 (PARI) rows_upto(n) = my(A, v1); v1 = vector(n+1, i, vector(i, j, 0)); v1[1][1] = 1; for(i=1, n, v1[i+1][1] = 1; v1[i+1][i+1] = 1); for(i=2, n, for(j=1, i-1, A = (i+j+1)%2; v1[i+1][j+1] = 2*sum(k=0, (i-j-1)\2, v1[i-j+1][2*k+A+1]*v1[j+2*k+A+1][j]))); v1 \\ _Mikhail Kurkov_, Aug 27 2025
%Y A078121 Cf. A002577, A016131, A078122, A089177.
%K A078121 nonn,tabl,changed
%O A078121 0,5
%A A078121 _Paul D. Hanna_, Nov 18 2002