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 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. 24
 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, 1, 1, 1828, 8148, 10816, 5456, 1024, 64, 1, 1, 27338, 167568, 274856, 174336, 43680, 4096, 128, 1, 1, 692004, 5866452, 11622976, 8909648, 2794496, 349504, 16384, 256, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 0..80, flattened FORMULA 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). 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<=k0 (conjecture). EXAMPLE The square of the matrix is the same matrix excluding the first row and column: [1, 0, 0, 0, 0]^2 = [ 1, 0, 0, 0, 0] [1, 1, 0, 0, 0] [ 2, 1, 0, 0, 0] [1, 2, 1, 0, 0] [ 4, 4, 1, 0, 0] [1, 4, 4, 1, 0] [10,16, 8, 1, 0] [1,10,16, 8, 1] [36,84,64,16, 1] MAPLE M:= proc(i, j) option remember; `if`(j=0 or i=j, 1, add(M(i-1, k)*M(k, j-1), k=0..i-1)) end: seq(seq(M(n, k), k=0..n), n=0..10); # Alois P. Heinz, Feb 27 2015 MATHEMATICA 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

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Last modified February 25 02:17 EST 2024. Contains 370308 sequences. (Running on oeis4.)